In a wild finish, the hosts fought back from a ten point deficit to almost pull off a remarkable victory.
It started with a try to Samu Kerevi, followed by an incredible 95 metres intercept by Mark Nawaqanitawase.
In the end, there was one twist remaining as poor discipline came back to haunt the Wallabies, with the count finishing 14-10.
Michael Cheika's side launched one final attack and number eight Juan Martin Gonzalez proved to be the hero.
The number eight dived over from short and whilst the Wallabies thought they had held him up, but the TMO found Gonzalez had dotted it down to break the hearts of Wallabies fans.
"We're all really disappointed mate. We put a lot into that," coach Eddie Jones said.
"We just couldn't put enough pressure on the opposition. Every time we got in position to put pressure, we either gave the ball back or didn't defend hard enough.
"So there's a couple of things we need to fix, which we can fix with a lot of hard work. And if we're prepared to do the hard work, we will get the fix. I’m sure the players will.”
The Wallabies replicated their strong start from last week as Len Ikitau barged over the line inside five minutes.
However, the incident left the centre clutching at his shoulder, forced from the field in the 20th minute. Ikitau went to hospital just after the second half for scans, with the severity to be confirmed.
Despite this, the hosts opened up a 10-0 lead off the back of a strong start from Will Skelton and the forward pack before Argentina worked their way into the contest.
They camped inside the Wallabies’ half, forcing them to make over 80 more tackles in the first 40 minutes. Dave Porecki and Allan Alaalatoa were among four Wallabies to record double-digit tackles in the first half as the Pumas wore them down.
Argentina’s relenting pressure eventually delivered them points through centre Jeronimo De La Fuente to level it up.
Replacement Carter Gordon nearly produced a magic try, scooping up a loose ball and racing away, only to be caught by Lucio Cinti 15 metres out.
Michael Cheika’s side continued to press the Wallabies’ line, leading to the yellow carding of Richie Arnold just before the break.
It allowed the Pumas to pick up where they left off and whilst Tom Wright and Quade Cooper were somehow able to hold Rodrigo Isgro up, they couldn’t stop captain Julian Montoya.
Needing a spark, Nic White stepped up and caught Argentina napping at scrum-time to level it up.
Emiliano Boffelli’s long range penalty put Argentina back in front and Mateo Carreras’s late try looked to have secured the win.
The Wallabies refused to die and got some great impact off the bench from the likes of Angus Bell and Rob Leota.
It took their International stars to set up the grandstand finish as Quade Cooper found Samu Kerevi.
With the game in the balance, Argentina attacked the hosts’ line and found themselves metres away from scoring.
Up stepped Nawaqanitawase.
The winger read the play perfectly, snatching the intercept and running 95 metres for the try in what appeared to be the hero play for the local, who had grown up just 20 minutes from the ground.
However, Argentina had one final chance to attack the Australian line when Cooper was pinged for a late tackle and made it count.
As the Wallabies looked to hold them out, number eight Gonzalez dove over the top and found the line.
The hosts protested, believing he was held up. However, referee Jaco Peyper ruled he got the ball to ground, sealing the emotional three-point win for the Pumas.
“To sum it up, gutted is the word that comes to mind,” captain James Slipper said after the match.
“I thought we turned up and started playing some good Rugby but we kept taking the pressure off the Argentinians and they kept coming back. Ill-discipline really hurt us again, yellow card and a couple penalties put us on the wrong side of the field.
“You just can’t win Test matches playing that sort of Rugby.”
[Tries: Ikitau, White, Kerevi, Nawaqanitawase; Cons: Cooper 4; Pen: Cooper] Argentina 34 [Tries: De La Fuente, Montoya, Carreras, Martin Gonzalez; Cons: Boffelli 4; Pens: Boffelli 2]
Australia: 15 Tom Wright, 14 Mark Nawaqanitawase, 13 Len Ikitau, 12 Samu Kerevi, 11 Marika Koroibete, 10 Quade Cooper, 9 Nic White, 8 Rob Valetini, 7 Fraser McReight, 6 Jed Holloway, 5 Will Skelton, 4 Richie Arnold, 3 Allan Alaalatoa, 2 David Porecki, 1 James Slipper (captain).
Replacements: 16 Jordan Uelese, 17 Angus Bell, 18 Pone Fa’amausili, 19 Matt Philip, 20 Rob Leota, 21 Josh Kemeny, 22 Tate McDermott, 23 Carter Gordon.
Argentina: 15 Emiliano Boffelli, 14 Rodrigo Isgro, 13 Lucio Cinti, 12 Jeronimo De La Fuente, 11 Mateo Carreras, 10 Santiago Carreras, 9 Gonzalo Bertranou, 8 Juan Martin Gonzalez, 7 Santiago Grondona, 6 Pablo Matera, 5 Tomas Lavanini, 4 Matias Alemanno, 3 Francisco Gomez Kodela, 2 Julian Montoya (captain), 1 Thomas Gallo.
Replacements: 16 Agustin Creevy, 17 Hahuel Tetaz Chaparro, 18 Eduardo Bello, 19 Lucas Paulos, 20 Rodrigo Bruni, 21 Lautaro Bazan Velez, 22 Nicolas Sanchez, 23 Matias Moroni.
Referee: Jaco Peyper (South Africa)
Assistant referees: Paul Williams (New Zealand), James Doleman (New Zealand)
TMO: Marius Jonker (South Africa)