The Russian third seed, who was pushed by Italian Karin Knapp in searing heat during a three-and-a-half-hour second-round match, had more troubles on her serve.
Sharapova dictated the contest against Cornet from the back of the court for the most part, winning 6-1 7-6 (8-6) in one hour and 51 minutes on a cooler Melbourne day.
The four-time grand slam champion is into the fourth round at Melbourne Park for the fourth straight time, but will need to improve on serve after producing eight more to take her tournament-high tally to 29.
She will face Slovakian 20th seed Dominika Cibulkova, who thrashed Spain's Carla Suarez Navarro 6-1 6-0.
A slow start by Cornet helped Sharapova to a 4-0 lead and the first set was over in 32 minutes when the Frenchwoman committed her ninth unforced error.
Cornet won just seven points on the Sharapova serve during the set, with the 2008 champion pushing her well behind the baseline.
But Cornet fought hard to start the second, recovering from 0-40 in the third game to hold for 2-1.
Cornet broke in the next game with a wonderful drop volley, but was broken to love – completed with a double fault.
Sharapova lifted a gear and Cornet had no answers, until the Russian struggled to close after taking a 5-3 lead.
Cornet saved a match point in the ninth game and levelled at 5-5 thanks to Sharapova's eighth double fault.
The pair traded breaks again, Sharapova saved a set point in the tie-break before Cornet sent a forehand long to end a 76-minute set.