Seeded third, Azarenka advanced despite being broken six times. She blew an easy overhead to lose the first game of the final set, then swept the last six games thanks to relentless baseline play that wore Cornet down.
American Sloane Stephens, seeded No. 17, reached the fourth round for the second consecutive year, beating Marina Erakovic 6-4, 6-7 (5), 6-3.
No. 13 Kei Nishikori became the first Japanese man in 75 years to reach the fourth round at Roland Garros, beating No. 24 Benoit Paire of France 6-3, 6-7 (3), 6-4, 6-1.
The center court stands were half-empty on a cloudy, mild day for Azarenka's match. Crowds will likely be bigger for her next week, and so will the stakes.
"Center court is something that motivates me every time I walk on court," she told the crowd after her victory, "and I can't wait to be back here."
Azarenka has won the Australian Open each of the past two years. Roland Garros is the only major tournament where she has yet to reach the semifinal.
Cornet, seeded 31st, stumbled and fell at the start of the second set and was slow to rise. She appeared unhurt but lost her composure as the match slipped away, screaming at herself between points and once spiking two balls in anger.
Over on court 1, Stephens won the last three games against Erakovic, who was penalized a point in the third set for getting coached.
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 reaching her first Grand Slam semifinal at the Australian Open this January, beating Williams to get there.
Erakovic was the first New Zealand woman ever to make the third round in Paris.
Like Erakovic, Paire was docked a point for being coached -- and at a critical time. He had a set point at 5-4 in the second set when he was given the penalty, which made the score deuce.
Paire won the set anyway but fell behind to stay when he dropped serve in the last game of the third set.
Nishikori broke serve eight times and benefited from nine double-faults by Paire. He became the first Japanese man to reach the round of 16 at Roland Garros since Fumiteru Nakano in 1938.