Ben Youngs came off the bench to score the only try of Saturday's Pool B match after 68 minutes and Jonny Wilkinson's conversion nudged England into the lead for the first time.
England's discipline was poor and they were second best for much of the game.
Had the Pumas not missed a total of six penalty shots and a drop-goal, they could have been out of touch.
Argentina's points came from two Martin Rodriguez penalties and one from Felipe Contepomi, the influential fly-half who was forced off injured in the first half.
Wilkinson was equally poor in front of goal, sending five penalty shots wide, four of them in the second half as England sought a way back into the game.
But it was Youngs who provided England's 'get out of jail free' card.
The Leicester scrum-half, making his first appearance since undergoing knee surgery in the summer, scampered over from the base of a ruck to seal England the win.
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The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
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