The Exiles battled all the way to overcome the Chiefs, who qualified for the Heineken Cup by finishing fifth last season.
Trailing 14-0 after 11 minutes, they hit back to finally snatch victory when a tremendous effort from their pack ended with Jackson driving over after 78 minutes to leave Jewell, who had come on for Gordon Ross six minutes earlier, a straightforward conversion.
The Exiles' earlier scores had come from a penalty try and a 70-yard interception try by left-wing Nick Scott. Former Scotland fly-half Ross landed two penalties and a conversion.
Full-back Luke Arscott, winger Matt Jess and number eight Richard Baxter scored Exeter's tries. Fly-half Ignacio Mieres landed three conversions with his second-half replacement Gareth Steenson kicking a penalty.
Welsh, promoted from the Championship last season, had found out the hard way that you get punished for your mistakes in the Premiership as they lost 38-13 to Leicester and 40-3 at Harlequins in their opening two Premiership matches,
Exeter made them pay again after only five minutes with the opening try from Luke Arscott, whose brother Tom lined up at full-back for the Exiles.
With Ross probing for openings, Welsh forced a five-metre scrum when Luke Arscott was tackled behind his own try-line.
But a sloppy pass from the base of the scrum by Tyson Keats presented winger Ian Whitten with the chance to intercept and run 70 metres before off-loading to send Arscott on a clear run to the line.
Six minutes later poor defensive play in midfield allowed powerful Fijian centre Sireli Naqelevuki the opportunity to power his way past three home players before sending Jess in for the second try. And with Mieres converting both, Exeter were 14-0 ahead.
But the Welsh pack hauled the side back into the game with a classic piece of forward play.
Their classic rolling maul - something of a forgotten art in modern rugby - had the Chiefs back-pedalling frantically.
Referee Dave Pearson almost immediately signalled he had penalised Exeter but allowed Welsh the advantage and, with the visitors transgressing again in a desperate bid to halt the momentum, the official awarded a penalty try which Ross converted.
Scott got Welsh on the board once more when he anticipated hooker Chris Whitehead's pass inside his own 22 to turn defence into attack and outpace his pursuers on a 70-metre run for a try in the left corner, although Ross missed the conversion.
Ross again missed the opportunity to kick the Exiles ahead in the 32nd minute after Exeter flanker Tom Johnson was penalised for going straight through a ruck.
But Ross made no mistake five minutes later, kicking Welsh 15-14 in front after Exeter were penalised for bringing down a scrum in front of their own posts.
Exeter upped the intensity at the start of the second half and a series of forward drives ended with Baxter driving over from close range for their third try, converted by Mieres.
Two minutes later, Ross cut the deficit to three points with his second penalty from 40 metres but Mieres missed a fairly simple chance to cancel it out in the 56th minute.
Steenson, brought on for Mieres in the 57th minute, kicked Exeter 24-18 ahead with his first penalty shot seven minutes later.
But Welsh mounted one final assault and a series of four powerful forward surges towards the Exeter line ended with Jackson breaching the Chiefs' defence to set up Jewell's winning kick.