Andy Goode looked to have secured victory for the Warriors just two minutes earlier after Gloucester skipper Jim Hamilton recklessly pushed Worcester lock James Percival at a lineout.
But Worcester could not hold on, and Burns struck with the final kick of an otherwise dire contest to secure a draw.
Gloucester centre Henry Trinder and Worcester replacement prop James Currie scored second-half tries, while Goode slotted two penalties and his team-mate Joe Carlisle a penalty and conversion.
Burns, who went on in the second period, slotted a penalty and conversion from his two kicks at goal and there was a penalty double from Billy Twelvetrees, but neither side covered themselves in glory.
Conditions were ideal for a free-flowing spectacle, yet both teams' repeated failure to master basic handling skills helped render the game largely a non-event.
Scotland wing Nikki Walker made his Worcester debut after moving to Sixways from the Ospreys earlier this year, while there were first Gloucester starts for wing Shane Monahan - who replaced an injured Charlie Sharples - and Australian hooker Huia Edmonds.
Twelvetrees found his range inside the opening two minutes, landing a 50-metre penalty that served notice to the Warriors he meant business.
Goode slotted an equalising strike 10 minutes later, although only after he required lengthy treatment following a crunching tackle by Gloucester scrum-half Dan Robson.
Despite a perfect playing surface, both sides struggled to string together meaningful attacking phases and Gloucester regained the lead through another Twelvetrees penalty.
Goode, who stayed on in an attempt to run off his knock, missed two penalties but Worcester should have scored from their first notable attack.
Former England forward Chris Jones shrugged off a couple of weak Gloucester challenges to power towards the line, yet visiting full-back Jonny May executed a stunning tackle that ensured Gloucester retained their lead.
The Warriors attempted to free Samoan wing speedster David Lemi at every opportunity, and although Gloucester marshalled him well, Worcester looked a more threatening attacking unit.
But Goode did not help his team's cause when he was sin-binned after taking out Trinder off the ball, and the visitors piled on pressure as half-time approached.
Wing James Simpson-Daniel was guilty of a woeful blunder when he failed to gather possession with Worcester's line at his mercy, yet 14-man Worcester somehow held out under intense pressure, trooping off just three points adrift.
Carlisle took over kicking duties while Goode served his 10-minute spell on the sidelines, and an angled penalty success made it 6-6 in a game that badly needed some inspiration.
Gloucester boss Nigel Davies made his first substitutions just 13 minutes into the second period, one of which included Burns replacing Mike Tindall and Twelvetrees reverting to his more familiar inside centre position.
Burns, who scored 39 points in Gloucester's opening two league games against Northampton and London Irish, looked to make an immediate impression on the contest.
He mixed his kicking game well but errors abounded around him until Gloucester struck with the game's first try 14 minutes from time.
Monahan made it possible through a powerful touchline burst and, as Worcester defenders desperately scrambled back, the ex-Rotherham player's well-timed pass sent Trinder sprinting over.
Burns converted but Worcester stirred to set up a gripping finale the game hardly deserved when Currie ploughed over for a try that Carlisle improved.
Gloucester found themselves stretched defensively for the first time and Worcester entered the final five minutes pushing hard to turn territorial dominance into points.
The victory they coveted looked to have arrived deep into injury time when Goode kept his nerve under intense pressure, but then Warriors' front-row infringed and Burns made it 16-16 with the game's final kick.
However, after kicking two penalties in quick succession to level proceedings, the 31-year-old was off target with a last-gasp drop goal in front of the posts.
Another England fly-half, Toby Flood, kicked the three penalties that had seemingly put Leicester on course for victory but the Tigers pivot was also off-target with a drop-goal attempt with the last kick of the game as the points were shared.
The place-kicking battle started early, with Flood successful with two penalties inside the first six minutes.
Sarries barely got out of their own half early on but slowly came into the match and could have reduced the deficit on 10 minutes.
But in a sign of what was to follow in a dismal first half for Farrell, making his first start of the season, the 20-year-old dragged his effort wide - the first of four misses from five attempts.
Leicester skipper Geordan Murphy was forced off through injury after 14 minutes, replaced by Matt Smith, before Farrell failed to take advantage with his next penalty attempt - dragging his effort off target from 40 metres.
The Tigers slowly took a grip on the game and enjoyed an impressive spell of possession, Vereniki Goneva and Scott Hamilton nearly linking up to produce the first try but for the timely intervention of Alex Goode on the left wing.
Despite sustained pressure they had to settle for a penalty, Flood extending the lead to nine with his third success on the half-hour mark.
Farrell finally got himself off the mark with his fourth penalty attempt on 34 minutes but he was wayward again with the last kick of the first half to leave the margin at 9-3.
Sarries kept the faith with their misfiring fly-half in the early stages of the second half but Hodgson joined the fray in the 51st minute and seconds later succeeded where Farrell had failed.
The former Sale number 10 - who retired from international duty in the summer - kicked Sarries to victory against his old club last week and he found his range immediately after Tigers were penalised for coming in at the side.
The scores were level before the hour mark when Hodgson made it two from two. Julian Salvi did not roll away quickly enough and the 31-year-old duly slotted the penalty to set up a grandstand finish.
Leicester had the next chance to restore their lead on 68 minutes but Flood pulled his kickable penalty attempt wide for his first miss of the day at a most inopportune moment.
Sarries wasted a chance on the counter-attack after some quick thinking by Goode, Richard Wigglesworth knocking on with a gap looking to have opened up.
A Saracens turnover and a kick ahead from Hodgson forced Leicester to concede a line-out within metres of their line heading into the last five minutes.
Safe line-out ball allowed Sarries to set-up the replacement fly-half with a drop-goal attempt in front of the posts but Hodgson sliced it wide.
Leicester got themselves down the other end of the field to give Flood his chance to be the hero but his effort was just wide as the points were shared.
Stephen Jones landed 23 points on his Wasps debut to help his new club to a 43-14 victory over London Irish - their first league win of the season at the third attempt.
The 34-year-old Welshman, with 104 caps for his country, produced a perfect kicking performance, slotting five penalties and also converting two tries from Tom Varndell and one each from scrum-half Joe Simpson and replacement prop Zak Taulafo.
It provided a bonus-point winning start for Wasps' new ownership consortium who took control of the club from former owner Steve Hayes this week.
Irish replied with a try from England Under-20 winger Marland Yarde and three penalties from fly-half Steven Shingler.
But they suffered their third successive defeat of the new campaign, in which they have now conceded 123 points.
Jones, replacing fellow-Welshman Nick Robinson after missing the first two matches with a back spasm, opened his Wasps account with a fifth-minute penalty.
Irish suffered another blow when prop Leo Halavatau limped off in the 10th minute to be replaced by John Ryan, on loan from Munster.
Four minutes later they missed out on the opening try when flanker Ofisa Treviranus knocked on as he crashed his way over the line.
But they drew level in the 15th minute with a Shingler penalty after the Wasps scrum was penalised.
Irish grabbed the opening touchdown after 25 minutes when Yarde scored a try out of nothing.
There seemed little danger when centres Sailosi Tagicakibau and Jonathan Joseph combined to supply Yarde, but the winger showed the ball, jinked and produced a burst of sheer pace which left four defenders standing before he crossed in the corner.
With Shingler off the pitch receiving treatment for a blood injury, full-back Tom Homer took over the kicking duties but missed the conversion.
Wasps were denied a try in the 28th minute when they were brought back for an earlier forward pass after full-back Hugh Southwell sent Varndell racing over in the left corner.
Handling errors let the home side down as they tried to over-elaborate but they forced two penalties, both landed by Jones - the second from just inside the Irish half - to go 9-8 ahead.
Shingler restored the Irish lead with a superbly-struck last kick of the first half after the Wasps scrum were penalised a metre inside their own half.
Taulafo replaced Fabio Staibano at half-time.
The second half was only a minute old when Jones kicked his fourth penalty as Wasps regained the lead after Irish offended at the breakdown.
They forged further ahead when Varndell went over in the left corner for a well-worked try. Right wing Christian Wade split the Irish defence, Southwell again provided the final pass and Varndell supplied the finish.
Shingler kicked his third penalty to cut the Irish deficit to five points.
But Wasps, who introduced England back-row man James Haskell from the bench, made victory an inevitability when Simpson took the ball from the base of a ruck and darted in for their second try, converted by Jones.
The Welshman landed his fifth penalty before Wasps drove Taulafo, scorer of a hat-trick of tries for Wasps in the A team game against Irish last Monday, over in the left corner two minutes from time.
But Wasps had not finished. Simpson burst 50 metres from inside his own half before passing to Varndell who raced round behind the posts to touch down. Jones rounded off a hugely impressive debut by converting the final two tries.
Reigning champions Harlequins made it three wins from three with a hard-earned 37-14 victory over battling Sale at The Stoop.
The final scoreline was harsh on the Sharks who competed throughout but the Quins were always the more creative side.
Harlequins made a blistering start. They immediately gained possession from the kick-off to produce a two-minute period of pressure, which ended with Ugo Monye being driven over for a try which Nick Evans converted.
Minutes later, Sale had their first chance for points but Danny Cipriani badly missed a 35-metre penalty attempt from almost straight in front of the posts.
When Mike Brown was penalised at a ruck 40 metres out, Cipriani was given another penalty chance and this time he made no mistake with his kick.
The opening quarter was frenetic with the hosts trying to exploit the ideal playing conditions by running the ball at every opportunity. However careless handling cost them at crucial stages, allowing the visitors to remain in touch.
Sale had another chance to reduce the deficit but Cipriani was narrowly wide with a penalty attempt from just inside his own half.
Against the run of play the visitors took the lead after 20 minutes.
A clever pass from Cipriani allowed Sam Tuitupou to make a big hole in the home defence. The ball was recycled and Richie Vernon raced beyond the cover defence for a try. Cipriani's conversion from the touchline rebounded back off the post.
The Sharks became more competitive but it was Harlequins who regained the lead six minutes before the interval with another excellent try.
They won quick ball from a lineout 40 metres out, Danny Care's swift pass sent Evans sailing though a huge gap in the visitors' defence and Jordan Turner-Hall was on hand to collect the scoring pass.
Evans converted to give the Quins a deserved 14-8 interval lead.
An excellent take-and-kick from Evans put Harlequins on the attack early in the second half and when Sale were penalised, the outside-half kicked a 30-metre penalty to extend his side's advantage.
Evans then saved the home side with a try-line tackle before kicking another penalty after the Sale pack infringed when under severe pressure at the set scrum close to their line.
Sale responded with a second penalty from Cipriani to be trailing 20-11 at the end of the third quarter.
Strong bursts from replacement Maurie Fa'asavalu and Monye gained the hosts a position in the Sharks 22 and once again Sale were penalised to allow Evans to kick his third penalty.
Cipriani gave Sale a glimmer of hope with his third penalty with 10 minutes remaining but that was his last slice of action as he was replaced by Nick Macleod.
With six minutes to go Harlequins effectively sealed their victory.
Mike Brown produced one of his trademark elusive bursts to set up the platform from which George Robson crashed over from short range for their third try which Evans converted.
Brown's late effort put gloss on the win.