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The 23-year-old Australian Open champion reeled off the last six games of the match and goes on to face either French 13th seed Marion Bartoli or Italy's 2010 champion Francesca Schiavone for a quarter-final spot.
"Alize is a terrific girl. It was a difficult match and she played really well, but playing on the centre court really motivates me," said Azarenka.
In a match which featured 15 breaks of serve, it was the 23-year-old Frenchwoman, fresh from a third career title in Strasbourg on the eve of Roland Garros, who took the first set, courtesy of a big forehand off a second serve.
Cornet took a tumble in the opening game of the second set and grimaced in pain again, feeling her upper right thigh in the eighth game as Azarenka went on to level the tie.
By that stage, Azarenka, who has never got beyond the quarter-finals in Paris, had already sent down 23 unforced errors and eight double faults.
Cornet was ahead 1-0 with a break in the decider before Azarenka pulled away, winning six games on the bounce to take victory after 2hr 22min.
Meanwhile, in other women's third round action, Sloane Stephens of the United States also reached fourth round, for the second consecutive year, with a 6-4, 6-7 (5), 6-3 victory over 92nd-ranked Marina Erakovic, who was penalized a point in the third set for getting coached.
The 17th-seeded Stephens won the last three games against Erakovic, the first woman from New Zealand to get to the third round in Paris.
A year ago, Stephens became the first U.S. teenager to reach the fourth round at Roland Garros since Serena Williams in 2001. She followed that up by getting to her first Grand Slam semi-final at the Australian Open this January, beating Williams to get there.
Now 20, Stephens will face defending champion Maria Sharapova or Zheng Jie for a spot in the French Open quarter-finals.
Stephens was three points away from closing out the 2h17min match against Erakovic much earlier, ahead 4-2 in the second-set tiebreaker. But Erakovic took five of the next six points to force a third set.
Stephens then faced two break points in each of her next two service games, but saved all four to lead 2-1.
At the ensuing change-over, the chair umpire announced Erakovic would be docked a point for coaching, which is not allowed during Grand Slam matches.
She argued and shook her head, and then went out and got broken to trail 3-1. But up a break against a flustered opponent, Stephens got broken right back when she put a forehand into the net. Erakovic held in the next game, making it 3-all.
But Stephens wouldn't drop another game.
French Open results on Saturday (x denotes seeded player):
Men:
Third round:
Kei Nishikori (JPN x13) bt Benoit Paire (FRA x24) 6-3, 6-7 (3/7), 6-4, 6-1
Women:
Third round:
Victoria Azarenka (BLR x3) bt Alize Cornet (FRAx31) 4-6, 6-3, 6-1
Sloane Stephens (USA x17) bt Marina Erakovic (NZL) 6-4, 6-7 (5/7), 6-3
