Stunned by a feisty Pumas outfit playing their first match on Australian soil since the 2003 World Cup, the Wallabies looked set for a shock defeat after two Argentinian tries had them behind 19-6 midway through the second period.
However, the quickfire double only served to spark the home side into action, with tries from Pat McCabe and man-of-the-match Digby Ioane sparing the Wallabies from embarrassment.
But the Pumas looked surprisingly good value for their lead on 50 minutes, after Tomas Leonardi charged down a Quade Cooper kick and then crossed for possibly the easiest five points of his career, with the ball bobbing up perfectly for him.
Then two minutes later, the Pumas tore Australia up on the right side with a move that began from well inside their own half and finished with tryscorer Julio Farias Cabello mobbed by ecstatic team-mates.
Juan Martin Fernandez, the Argentina captain, missed both conversions but made no mistake with his 57th-minute penalty goal, which put the visitors in the box seat for an improbable win.
But McCabe had the reply for the Wallabies on the hour mark, bursting through to put the ball down near the posts on an assist from the influential Cooper.
Then Nick Phipps opened the door for Australia's follow-up with a blindside run on the left that put Ioane in the clear and over the line, releasing the pressure valve for under-fire coach Robbie Deans.
Berrick Barnes, the full-back experiment in place of Kurtley Beale, added the extras on both occasions to give the Wallabies a 20-19 edge that the tiring Pumas could not recover from.
The versatile Queenslander added one more penalty late on to confirm the win.
Earlier, McCabe had saved a certain Pumas try from winger Gonzalo Camacho, who tore away at pace after stripping the ball off Ioane early in the first half.
Argentina's aggressive opening would have paid off there and then were it not for McCabe's important intervention, although referee Wayne Barnes saw it worthy of a 10-minute sin-binning.
Either side of that close call, Australia had two of their own - Barnes hit the right post twice from penalty attempts.
Despite weeks of warning that the Pumas meant business in their first appearance in the expanded Rugby Championship, the sloppy Wallabies could not have looked any less ready.
Their hiccups continued just after the 20-minute mark. Taking the ball from left to right, they stretched Argentina with ease and it fell to veteran Radike Samo to finish off a flowing move.
But he dropped the ball as he went over in the right corner, with video referee Matt Goddard handing the Pumas another let-off.
Barnes straightened up in front of goal in the 26th minute to break the drought with a penalty from close range, but Hernandez replied for Argentina shortly after an Ioane offside.
Five minutes from the break, Australia were thwarted again as an inviting ball from Cooper allowed Sharpe to break and head straight to the line - only for him to be rolled on his back by Camacho at the vital moment, robbing him of a try in front of his hometown crowd.