But the win looked to have come at a huge cost with playmaker Quade Cooper believed to have ruptured an Achilles tendon, an injury which could rule him out of next year's Rugby World Cup and possibly end the 34-year-old's career.
Centre Len Ikitau crossed for the Wallabies' fifth try five minutes after full-time in Mendoza to snare a bonus point as they rallied from a 19-10 halftime deficit and piled on 31 points after the break.
The Australians entered the Test without talismanic captain Hooper, who withdrew from the tour the day before the game for mental health reasons.
Stand-in skipper James Slipper said they wanted to make Hooper, who had already flown home to Australia, proud.
"It's a real proud moment for me and also the team because we really wanted to put in an effort that our fans could be proud of and also our captain Hoops (Hoops)," Slipper said in a post-match interview.
"We were thinking about him all day and playing for him."
The Wallabies lost another of their most experienced players when 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 strain however the injury appeared far more serious as he was assisted from the field in obvious pain.
But the Wallabies regrouped with Hooper's replacement Fraser McReight dragged across the line for his first Test try 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 within two points of Argentina.
With Argentine lock Matias Alemanno yellow-carded, Cooper's replacement Reece Hodge banged over a long-range penalty for the Wallabies to take the lead, 27-26, for the first time in the match.
They extended that in the 70th minute when 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.
The opening half was a frustrating stop-start affair with a 10-3 penalty count against the Wallabies.
The Australians dominated possession and territory but errors and poor discipline 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 and got out to a 13-3 lead.
The Wallabies countered through winger Jordan Petaia but kept the home side in the game through sharp-shooter Emiliano Boffelli.
The Australians got on the right side of Adamson in the second half, while the Pumas own discipline fell apart as they incurred 11 second-half penalties which cruelled their cause.
Wallabies coach Dave Rennie praised his team's resilience and poise after the late withdrawal of Hooper and then losing Cooper.
"We created first-half opportunities that we didn't take but I thought we were far more clinical and applied a lot more pressure (in the second half)," Rennie told Stan Sport.
"I was really happy with the way we finished.
"We've got more in us but I was really happy with the character.
"We had a couple of late changes and losing a key guy in the game.
"The thing about this group, a lot of character and courage. and they stood up."
The Wallabies face Argentina again next weekend in San Juan.
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