The Wallabies had won 15 of the past 16 matches between the two teams, including the past six, and had not been beaten by the Pumas on home soil since 1983.
But, on a day that had already seen South Africa shock New Zealand in Wellington, a second upset came to pass as head coach Mario Ledesma, a former member of Michael Cheika's Australia backroom team, masterminded a famous victory.
Argentina has now won two matches in the Rugby Championship for the first time, while more questions will be asked of Cheika and an Australia side that has now lost three of four matches and only scraped past South Africa last time out.
Emiliano Boffelli nailed a penalty from inside his own half to put Argentina on the board first, but sensational hands from first Dane Haylett-Petty and then Reece Hodge sent Will Genia over.
Argentina responded in kind with Nicolas Sanchez, who last week became the Pumas' all-time leading points scorer, receiving Jeronimo De La Fuente's pop pass.
The action-packed start continued when Israel Folau found a gap to score a fine solo try and Argentina was denied a second five pointer when Ramiro Moyano was deemed in touch when offloading to Sanchez.
But Argentina was ahead by the break when Pablo Matera's bulldozing run was supported by Marcos Kremer, who in turn offloaded to Bautista Delguy to race over.
The Pumas had another try disallowed early after the restart when Delguy was adjudged to have passed forward before Boffelli sprinted over the line.
A Sanchez penalty stretched the advantage, but Australia moved within a point when the ball was worked right from a rolling maul for Haylett-Petty to dot down in the right corner.
Sanchez once again eased the tension with a nerve-settling three pointer, but it took a huge tackle from Tomas Lavanini to deny Folau with the last play of the game.