Stosur took a 3-0 lead in the second set and held the rest of the way.
The ball kissed the baseline for a winner, and Stosur waved in apology to Mladenovic.
Sam Stosur, Francesca Schiavone, Victoria Azarenka and Elena Dementieva also won but Dinara Safina was beaten.
Australia's Samantha Stosur, the No. 5 seed, beat Carla Suarez Navarro, from Spain, 6-1 6-3.
Australia's Sam Stosur is the fifth seed, with four-time Wimbledon champion Serena Williams seeded sixth.
"I desperately wanted to win out there today, " said eighth seed and 2006 fdoubles finalist Stosur.
Meanwhile, Samantha Stosur of Australia won her second title on the women's WTA Tour circuit on Sunday.
Sam Stosur rounded off Tuesday's play with a 6-4 6-4 win over French Open champion Francesca Schiavone.
After Sharapova and Kvitova, the highest-ranked player remaining in the last eight is U.S. Open champion Sam Stosur.
The No. 9-seeded Stosur never lost serve but needed nearly an hour to take the first set against Mladenovic.
Seventh seed, Stosur admitted the match against Robson, who is currently ranked 89th in the world, will be tough.
"I ran in thinking she would drop-shot, and she didn't drop-shot, " Stosur said.
Australian second seed Sam Stosur joined Serena in the third round after defeating American wildcard Jamie Hampton 6-0 7-5.
Unseeded Italian Alberta Brianti lies in waits for Stosur after she upset 21st seed Sabine Lisicki 2-6 6-4 6-4.
Sharapova triumphed 6-0 6-3 against Stosur, surrendering just five points in an opening set that lasted only 21 minutes.
Bob Bryan and Stosur showed impressive resilience to battle back from a first set in which they were thoroughly outplayed.
Kvitova can put the Czechs into the semifinals by beating former U.S. Open champion Stosur in Sunday's opening reverse singles.
Australian sixth seed Samantha Stosur , the beaten finalist in 2010, breezed past Russian No. 27 Nadia Petrova 6-3 6-3.
Fourth seed Serena took just 44 minutes to dispatch Australia's Samantha Stosur, reeling off the last 10 games to love.
The 2011 Wimbledon champion saved a match-point at 5-4 down in the second set against that year's U.S. Open victor Stosur.
Meanwhile at the WTA event in Charleston, South Carolina, Serena Williams crushed Samantha Stosur 6-1 6-1 to reach the Sunday's final.
The tournament's fifth seed Nicole Vaidisova had a tough opener against Samantha Stosur but came through the test 3-6 6-4 6-4.
Meanwhile, in the women's event, Australia's Samantha Stosur set up a clash with Venus Williams after defeating Swiss Patty Schnyder 7-6 6-2.
Australia's top hope Samantha Stosur (13) will avoid a seed in the next round after the Queenslander beat Kristina Barrois 7-5 6-3.
Stosur's calf injury gifted Germany's fourth seed Angelique Kerber a place in the last four where she will now play Caroline Wozniacki.
The women's quarterfinals were blighted by injury after defending champion Victoria Azarenka and seventh seed Sam Stosur both pulled out with injury.
The Frenchwoman upset Venus in Sunday's final in Stanford, while world No. 2 Serena was beaten by Samantha Stosur in the quarterfinals.
Australian eighth seed Samantha Stosur defeated France's Julie Coin 6-1 7-6 (7-4) to set up a clash with Russian 25th seed Anastasia Pavlyuchenkova.
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