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."
"I definitely need to step it up -- every player will need to raise their game," she added, contemplating the tournament's second week.
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.
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.
In contrast, the Russian has reached eight major finals and won four, with her last success coming at the French Open in 2012.
The third seed got the match underway serving to love and made an early breakthrough in a tight second game highlighted by a series of entertaining baseline rallies, breaking to go 2-0 ahead when Cornet hooked a forehand long.
Sharapova's serve was finding its mark and she comfortably held before her pinpoint ball placement forced Cornet into a series of errors that allowed her to go 4-0 in front.
But her serving issues returned in the fifth game when a double fault gave Cornet a second break point and she grabbed the opportunity.
Unperturbed, the Russian broke straight back to love and served out for the set in 32 minutes.
Sharapova threw away a chance to ram home her advantage in the third game of the second set, blowing three break points as Cornet fought a rearguard action.
The Frenchwoman, who partnered Jo-Wilfried Tsonga to Hopman Cup success earlier this month, made the most of her reprieve, breaking the third seed with a deft drop shot to take a 3-1 lead.
But she wasted the advantage when she let the Russian break back.
With the finish line in sight, Sharapova got another crucial break in the seventh game with a forehand down the line to go 4-3 in front.
Cornet saved a match point on her serve at 3-5 to keep the tie alive and Sharapova then failed to serve out for the match.
She paid the penalty with the showdown going to a tiebreak, in which she saved a Corner set point before finally getting the win to book her spot in the last 16.