Substitute Ben Donaldson knocked over his effort from close range with the final action of the match to see the Wallabies to a narrow victory in cold and rainy conditions at Estadio Jorge Luis Hirschi in La Plata.
Australia outscored their hosts two tries to one with Jake Gordon and Rob Valetini both going over and Noah Lolesio adding a penalty and two conversions.
Argentina's try was scored early by loose forward Juan Martin Gonzalez, while Santiago Carreras contributed 11 points with three penalties and a conversion.
The victory will be a major fillip for the Wallabies after being comprehensively beaten at home by South Africa in their opening two matches of this year's Southern Hemisphere championship, although they remain bottom of the standings at the halfway mark.
"(I'm) really proud," Australia coach Joe Schmidt said.
"We have a lot of work to do but we're pleased to get the four points and we're in the contest."
The Test in La Plata was always going to be a scrappy affair given the foul weather but it also saw many positives, mainly on defence, for both teams.
After seeing off more than 30 phases of attack from Australia after the start, it was Argentina who opened the scoring with a Carreras penalty in the 12th minute followed three minutes later by a pushover try from a lineout maul for Gonzalez
Argentina's defence were busier than their attack in the first half, however, and they held out under persistent attack from the visitors until the Wallabies eventually managed to prise them open with a 27th-minute try from scrumhalf Gordon after Tom Wright had burst through following some slick passing moves.
Argentina were 13-7 up at the break as Carreras put over a second penalty and the fly-half added a third five minutes after the break to extend the lead.
Australia reduced the deficit to two points in the 50th minute with a try from Valetini after Len Ikitau beat two defenders with a break and quick ball thereafter saw them work their way over the line.
As the rain lashed down, Lolesio put over a 59th penalty for Australia to go ahead for the first time but with 10 minutes left Argentina were back ahead after Tomas Albornoz slotted a penalty to make it 19-17.
In the closing stages, Australia had the attacking impetus and Argentina gave away a needless penalty right in front of their own posts with a minute remaining for infringing at the ruck.
Donaldson, who had come on a minute earlier for Lolesio, had the easiest of tasks to put the ball over as the hooter to indicate the end of the match blew.
The Test was the last for 39-year-old former Argentina captain Agustin Creevy, who came on for the last eight minutes, marking his 110th cap, and a rapturous reception from the crowd in his home town.
