Swedish captain Nilla Fischer opened the scoring in the 17th minute after the ball bounced off her shoulder following a corner kick, but France rallied with a header by Georges in the 29th and a close-range shot by Renard in the 39th.
France, making its Olympic debut, at least equals its best campaign in a major women's tournament. It also reached the semifinals of last year's World Cup. It will play Brazil or World Cup champion Japan in the Olympic semifinals.
"We showed today how dangerous our defenders can be on set pieces, this team is not only about our attackers," Georges said. "We were able to score even against big and strong defenders like the Swedes have."
France coach Bruno Bini said he couldn't have asked for more from his defenders.
"We played a creative match. It was perfect," he said. "At corners, we scared the Swedish defense, with Wendie and Laura really strong there. We didn't buckle against a really strong team."
Sweden also made it to the semifinals of last year's World Cup and beat France 2-1 in the third-place match.
"It feels empty," Swedish defender Annica Svensson said. "We played a good game and I don't think that France played a better game than us. That feels kind of disappointing."
Fischer nearly scored her second goal to even the match in the 72nd, but French goalkeeper Sarah Bouhaddi made an incredible point-blank save to keep France ahead. Bouhaddi dived to her right and barely tipped Fischer's close-range shot wide.
France came into the match at Hampden Park with the tournament's best attack, with eight goals scored, tied with the two-time defending champion United States.
But it was Sweden that got on the board first in a match in which both sides struggled to keep possession but created many scoring opportunities.
Fischer went for a corner cross into the penalty area, and the ball bounced off her left shoulder as she looked away, going straight into Bouhaddi's left upper corner.
The French tied it after a well-executed corner kick by Louisa Necib. Her shot curled dangerously toward the goal, and Sweden goalkeeper Hedvig Lindahl had to step back to stop it from going in. She tipped the ball away from the net, but it went into Georges' direction for an easy header into the far corner.
France's go-ahead goal came after a long free kick cross from the right side. Georges headed the ball across the area, and Renard scored with a firm shot from close range.
Necib, who controlled the match for France in midfield, had a 22nd-minute goal disallowed for offsides before the ball was crossed into the area. The French complained because it appeared the ball came off a defender, which would have allowed the goal to stand.
Necib also nearly scored with a 72nd-minute header that missed wide.
Swedish defender Linda Sembrant had some of the best chances for Sweden. Her header off a corner in the 53rd just missed wide of the far post.
Sweden had relinquished a two-goal lead in a 2-2 draw with Canada in its last group match.
France's semifinal appearance in last year's World Cup in Germany marked one of the team's top achievements in women's soccer. France eliminated England on penalties in the quarterfinals before losing 3-1 to the United States in the semis.
The Swedes have always been a top contender in women's soccer, but their last major triumph came at the 1984 European Championship. It just missed on a medal at the 2004 Athens Games, finishing fourth.
Sweden's best finish at the World Cup was second in 2003 in the United States. It was eliminated in the quarterfinals of the 2008 Beijing Games after a loss to Germany.
Hampden Park had a crowd of just less than 13,000 for its final Olympic match on a rainy afternoon in Glasgow.
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