The result at the Allianz Stadium was another heartbreaking loss for Wales, who were beaten in the second Test by a Mike Harris penalty after the hooter.
Wales also lost the first Test in Brisbane, despite only trailing by a point in the second half.
Barnes scored four first-half penalties, adding the winning fifth penalty in the 75th minute, while full-back Leigh Halfpenny kicked four of his own plus a conversion to complete Wales’s points tally.
Fittingly for a tense, well-matched series, the opening half was a cagey affair as Wales sought their first win against the Wallabies on Australian soil since 1969.
But the tourists, who could reasonably have edged the series 2-1, will head back to the northern hemisphere ruing the cheap penalties they gave away, especially in this final Test in Sydney.
Australia took the lead just four minutes into the clash, Barnes missing a drop goal before coming back to kick a penalty after Wales were caught offside.
The Welsh levelled up just four minutes later after Nathan Sharpe was penalised for charging in from the side.
But just as Wales looked to be finding their feet, a shocking period of play saw them not only give away another penalty immediately which Barnes duly converted, but also a second, albeit it in Australian territory, as their re-start failed to travel the required distance.
Luckily for the Welsh, however, Australia were also far from producing tight play as they conceded a penalty at the scrum five minutes later. Halfpenny took full advantage with an epic kick, drilling the ball through the centre of the posts from the half-way line.
The Wallabies regained the lead at the end of the first quarter, Wales penalised for not rolling away from the tackle, before a better series of moves from the tourists drew another penalty of their own when Tatafu Polota-Nau failed to roll away from a last-ditch tackle on Alex Cuthbert.
Wales did well to cope with the loss of talismanic skipper Sam Warburton, the forward departing the field after taking a knock to the head, on the half hour mark but Australia had the last word in the half as Barnes kicked them into the lead once again after Mike Phillips was caught offside.
The early stages of the second half looked set to follow in much the same vein as the first but as defensive legs tired, gaps opened up for both sides to try and take advantage of.
Australia were the first to provide some real excitement as the returning Kurtley Beale, generally poor for the Wallabies in his first match in a month, connected with Sitaleki Timani only to run straight into Halfpenny after putting his chip over the top to chase.
Next it was the turn of Wales as Cuthbert put a kick upfield for Justin Tipuric, the Welsh replacement just beaten to the loose ball by Digby Ioane.
However, the taste at running rugby seemed to inspire Wales and they finally did get their first, and only, try of the night six minutes later just after the hour to take the lead.
Phillips took the ball to within five metres of the line before popping the ball up for the ever-present Cuthbert. The Welsh winger was hauled down just short by the Wallabies but Ryan Jones was in place to dive over the line in between the posts.
Halfpenny added the conversion without trouble but, as is so often the way with southern hemisphere teams, Australia bounced straight back, scoring a try of their own just three minutes later.
After working the ball from one wing to the other, Horne received the ball from Will Genia with Beale outside him. The centre decided to go it alone, a move that nearly backfired as he only just held on to the ball, wedged between elbow and thigh, as he went down to the ground.
The TMO judged Horne to have had enough control of the ball to awarded the try, although the hosts failed to added the extra two points the lead as Barnes missed his only kick of the night with his conversion attempt.
Wales regained the lead with nine minutes remaining, Australia again being penalised at the breakdown (an area in which they struggled all day) and the metronomic Halfpenny, who has only missed one kick all series, doing his bit with the boot.
But Australia were not finished yet with Barnes recovering his kicking range in time to thread a penalty inside the right post four minutes later.
Wales had one last go at glory; substitute James Hook only being denied by the bounce of the ball as he comfortably beat a tired Barnes for pace, chasing a loose ball down the right wing.
Australia wound down the clock with solid possession and Wales will return home justifiably feeling that the 3-0 series scoreline is an unfair reflection of their performances on Australian soil.