Centre Len Ikitau crossed for their fifth try five minutes after full-time in Mendoza to snare a bonus point as the Wallabies piled on 31 points in the second half.
The Australians entered the Test without talismanic captain Michael Hooper, who withdrew from the tour the day before the game for mental health reasons.
And they lost another of their most experienced players when playmaker Cooper slipped on the turf while in attack seven minutes into the second half.
The 34-year-old had already missed the losing England series with a calf injury but appeared to grasp at his ankle in obvious pain, and was later be seen in a moon-boot.
But the Wallabies regrouped with Hooper's replacement Fraser McReight dragged across the line by barn-storming Test debutant Jed Holloway.
A penalty try in the 62nd minute awarded by whistle-happy Scottish referee Mike Adamson put the visitors to within two points.
Cooper's replacement Reece Hodge then banged over a long-range penalty for the Wallabies to take the lead, 27-26, for the first time in the match.
Hooker Folau Fainga'a, in his best performance in the gold jersey, picked up the ball off the back of a maul and charged over, with Hodge adding the extras.Â
The opening half was a frustrating stop-start affair with a 10-3 penalty count against the Wallabies, who trailed 19-10 at the break.
The Australians dominated possession and territory but errors and poor discipline again stopped them from maximising their opportunities.
Argentina, who are coached by former Wallabies mentor Michael Cheika, got off to their best possible start with Pablo Matera crossing in the sixth minute.
The Wallabies countered through winger Jordan Petaia but kept the home side in the game through sharp-shooter Emiliano Boffelli, who booting four penalties for the game.
6°C