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."
Sharapova will next play Dominika Cibulkova, who beat No. 16 Carla Suarez Navarro 6-1, 6-0 in 59 minutes. Suarez Navarro was clearly still fatigued from her 3-hour, second-round match in the extreme heat. She hit only two winners against Cibulkova.
"I finished the last match with pain. I tried to recover yesterday but it was not possible to play good today," she said after Saturday's defeat. "When you play with these players at this level, you need to be 90 percent perfect or 100 percent perfect. If you are less than this, you cannot play, you cannot be on court."
Stephens, 20, advanced to the fourth round with a 7-5, 6-4 win over Elina Svitolina of Ukraine.
Last year, Stephens made it to the fourth round or better at all four majors -- joining Serena Williams and Agnieszka Radwanska as the only players to achieve that feat. And Stephens, seeded 13th this year, beat Williams in last year's quarterfinals at Melbourne Park, one of only four losses for Williams in 2013.
Wozniacki now will have to wait until at least the French Open to claim an elusive first Grand Slam title. Muguruza Blanco won her first WTA tournament two weeks ago at the Hobart International.
The Associated Press contributed to this report.