Azarenka had all kinds of trouble early on in the match - after rallying from 4-1 down to push the first set to a tie-break, she watched on as an inspired Cornet kept cracking winners from everywhere to take that breaker handily, 7-2, and the two players were even through 2-all in the second set as well.
But that's when the No.2-seeded Azarenka pulled away, cutting down on her errors and staying in the rallies more with the No.26-seeded Cornet, closing it out in two hours and 40 minutes, 67(2) 63 62.
"I don't think I played my best tennis today for sure, but I have to give her credit - she played really well," Azarenka said afterwards. "I felt like I didn't take my opportunities in the first set, which was kind of a waste, and it gave her a lot of confidence. She really came up with great shots when she needed them, and she was hitting some big serves too. I've actually never seen her serve like that.
"I'm glad I could turn things around in the second and third set. That was when I was really finding my rhythm. And I felt much better in the end after the two hour mark, which I'm very happy about."
Next up for Azarenka is No.13 seed Ana Ivanovic, who also survived a marathon, just edging American qualifier Christina McHale, 46 75 64. McHale actually served for the match at 64 54, but Ivanovic got the break back quite handily and eventually moved through in a two-hour, 26-minute marathon.
"I played a great game to break her when she was serving for the match - that gave me confidence and from there I kept trying to move forward and be aggressive, even when I was making some mistakes," Ivanovic said. "And then in the third set I was serving very well. I'm just so happy to be through."
Ivanovic was asked if a tough win like this was better in any way than a simple straight set win.
"Sitting here now after I won, of course I'm going to say it's great, and that it challenged me and pushed me," Ivanovic said. "If I didn't win the match today, it would be hard to see it that way, of course.
"But you're not going to win 60 61 every match unless you're Serena. You have to battle out even when you're not really on top of your game. So now I have an opportunity to improve for my next match."
Azarenka leads Ivanovic in their head-to-head series, 4-2, and has won their last three meetings.
"Ana is a great player," Azarenka said. "She has really improved. I think especially this year she's been showing some consistent results. We had a great match in Carlsbad, which was my first tournament after coming back. She's definitely in great form and I'm sure she's motivated and pumped up now.
"It's going to be a good match. She's a great champion, as well. I'm looking forward to it."