Azarenka broke Meusburger's serve twice in the first four games and was rarely troubled in exactly 1 hour to advance to a fourth-round match against American Sloane Stephens.
Last year, Azarenka beat Stephens in the semifinals at Melbourne Park under acrimonious circumstances when Azarenka took a questionable medical timeout while trailing Stephens. When Azarenka returned, she broke Stephens' serve and went on to win the match.
"Sloane is a great player; she's improved so much from last year," the second-seeded Azarenka said Saturday.
For her part, Stephens said she doesn't get flustered as easily now and she's not overwhelmed by matches on the big stages. She's also learning to focus on herself and tune out distractions on the other side of the net.
"Last year has nothing to do with this year," she said. "It's a new match."
Her third-round match was played in high humidity but in temperatures of about 72 Fahrenheit, considerably cooler than the scorching 108-degree conditions she endured for 3 hours, 28 minutes in her second-round win over Karin Knapp two days previously. The tournament's extreme heat policy was enacted for the first time in five years during that match.
Again, though, Sharapova struggled to close out Saturday. She took 50 minutes between her first and last match points against Knapp, and needed almost a half-hour to finish off Cornet -- she missed a match point with a wayward backhand on the Frenchwoman's serve and then got broken twice while trying to serve out.
Sharapova is slowing finding her groove in her second tournament back after a prolonged break for a right shoulder injury. She had six double-faults and 29 of her total 35 unforced errors in the second set after breezing through the first against Cornet.
"After the last match, I'm just happy to get through this," Sharapova said. "Definitely need to step it up. I was lucky to get through the other day; now that I'm in the second week, I'm level."
The four-time major winner needed an ice bath after her second-round win but joked about needing a warm bath following her victory over Cornet.
"It's such a quick change," Sharapova said of the cooler conditions. "I think it's really welcome from all of us.
"Everyone that played a long match in those conditions is going to feel physically and emotionally tired, and that's the way it goes. You just have to find a way to get through it ... that's what I did."
The third-seeded Sharapova was still wearing ice vests and draping ice-filled towels over her shoulders in the changeovers Saturday.
No. 25 Cornet appeared to be laboring between points in the second set, spending time retreating to the shade and breathing deeply at certain stages. She had been clearly distressed after her second-round win in the heat, sobbing when she described the conditions as like "an oven."