The Wallabies conceded two tries in the first four minutes and trailed 31-7 at the break – before scoring five second-half tries to secure a 45-34 victory.
Television cameras showed Australia coach Michael Cheika, who has been under pressure after a poor run, giving an angry half-time team talk.
“It’s not really for public airing what I said,” Cheika said.
“I just wanted to say what I felt. I didn’t go down there with a plan of doing that or anything like that.
“I was obviously very unhappy with the first half and very happy with the second half – but happy doesn’t really make a difference at the end of the day.”
Australia’s win is the biggest comeback involving tier-one nations, surpassing the mark of Fiji, who came from 21 points down at half-time to beat Tonga 41-38 in 2010.
The largest comeback in international Test history came in 2007, when Chile trailed Uruguay 27-0 at the interval, only to win 35-34.
Victory also meant the Wallabies moved above the Pumas to finish third in the competition behind South Africa and winners New Zealand, who staged their own stunning comeback to beat the Springboks in Pretoria.
Australia captain Michael Hooper hit back with a try but the visitors conceded twice more before the break, Matias Orlando touching down and Santiago Gonzalez Iglesias scoring a superb solo effort after replacing injured fly-half Nicolas Sanchez.
After Cheika’s half-time intervention, the Wallabies dominated the start of the second half, with Izack Rodda, Israel Folau, Dane Haylett-Petty and David Pocock crossing and Bernard Foley converting all four tries to put the visitors 35-34 in front after 65 minutes.
Haylett-Petty added his second try moments later before Foley kicked his sole penalty of the game to round off a remarkable win.