The Pumas needed at least a draw from the clash to qualify, but they could not get on the score-sheet until after the 30-minute mark, when winger Juan Jose Imhoff went over for the first points of the game.
Argentina dominated possession in the first half, but they struggled to break a spirited Georgian defence, and captain Felipe Contepomi missed two penalty chances early on.
Georgia, meanwhile, were plagued by discipline and handling errors going forward, despite occasionally venturing into Argentina's 22 with pick-and-drive plays.
The Lelos' hard work paid off just before half-time when fly-half Lasha Khmaladze took advantage of a sloppy Argentina defence to go over for his first international try.
Fullback Malkhaz Urjukashvili converted to give Georgia a two-point lead going in to the second half.
Argentina re-took the lead after the break with two successful penalties from Contepomi, followed by a try and conversion from the captain with just over 10 minutes left.
Agustin Gosio darted down the wing with minutes remaining to score his fist international try, and centre Marcelo Bosch converted to make the final score 25-7.
The Pumas will go on to face New Zealand in a quarter-final clash on Sunday, October 9.