The four-time Grand Slam champion came through her match on Rod Laver Arena 6-1, 7-6 (8/6) in one hour 51 minutes, but Sharapova's killer instinct deserted her in the second set when she failed to serve out for the victory, forcing a tiebreak.
"She started playing better when I gave her some chances," said Sharapova, after setting up a clash with 20th-seeded Slovak Dominika Cibulkova for a place in the quarter-finals.
"After my last match I'm just glad to be through and have another day to recover."
Sharapova is still on the comeback trail from a shoulder injury and she was pushed hard in draining heat during her second-round match against Karin Knapp, eventually coming through a marathon third set that went 10-8.
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Despite those energy-sapping conditions, she showed few ill-effects on Saturday although in temperatures that were some 20 Celsius cooler, she was still using ice vests on the changeovers.
Getting past Sharapova was always going to be a big ask for Cornet. She has only beaten a top-10 player twice in her career and had only been beyond the third round once in 31 Grand Slams, at the 2000 Australian Open.