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The Pumas only needed one point to qualify after Scotland's heartbreaking late 16-12 loss to England on Saturday, but they made it heavy going finally wearing down the plucky Georgians three tries to one.
The 2007 semi-finalists were out of sorts in a muddling opening half trailing 7-5 at halftime before posting 20 unanswered points to win convincingly enough in the end.
"Georgia played really well in the first half, we couldn't score when we had to and they put us to the limit," said skipper Felipe Contepomi, who is nursing a rib cartilage injury.
"We tried to improve our game in the second half and thankfully we scored some tries and we won, but we have a lot to do in the next week."
Pumas coach Santiago Phelan while delighted to reach the last eight, was critical of his team's performance.
"The team didn't play well today, they were not structured and this meant they didn't achieve what they wanted to on the field. But it was important to hang in there and the game eventually came our way," Phelan said.
"We tried to take the game on but lacked precision and that led us into making errors. Maybe we lacked the focus we had last week (against Scotland)."
It was Argentina's second win over Georgia at the World Cup, having beaten them 33-3 in Lyon at the 2007 tournament.
"It's a shame because we played well and in the end they scored two tries. Maybe we were too tired in the end because of our last match against Romania four days ago, said Georgia prop David Zirakashvili.
"The final score does not reflect the way we have played."
The Argentineans struggled for any fluency in a mistake-ridden first half with Contepomi adding to the Pumas' problems by missing all his first three goal kicks.
But it was the Pumas who finally scored first eight minutes before the break.
Flanker Julio Farias Cabello gave winger Juan Imhoff some room on the left wing and he pushed off fullback Malkhaz Urjukashvili to score the opening points.
But the Georgians hit back minutes before the interval when right winger Alexander Todua kicked ahead for fly-half Lasha Khmaladze to regather and score a smart try.
Urjukashvili kicked Georgia in front with the conversion.
Contempomi, who was more involved in the second half, edged the Pumas in front with two penalties but he settled the outcome with a 69th minute try.
The Pumas swept downfield off a scrum with Lucas Gonzalez Amorosino and Horacio Agulla taking play to within five metres of the Georgian line.
Four swift passes to the right and Contepomi stepped inside a couple of tacklers to score the Pumas' second try and a 18-7 lead.
The Stade Francais playmaker was hauled off minutes later to ice his injured ribs and prepare for next weekend's quarter-final with the All Blacks.
Contepomi finished with 13 points to take his international points total to 586, just eight points off breaking Hugo Porta's Argentina record of 593.
Argentina grabbed their third try in the final minute though replacement back Agustin Gosio, who swooped on to a grubber from Imhoff to score in the right corner.
