The seventh-ranked Sharapova, who lost in last year's final to Kim Clijsters, won a challenge on match point to improve to 3-0 this year and 9-0 in her career against Stosur.
"I did many good things today that caused her a lot of trouble," Sharapova said. "I didn't give her much time to do the things she likes to do." Stosur is confident that she'll eventually beat Sharapova.
"It's possible to win," she said. "I've got to believe that, but I guess it's back to the drawing board." The 10th-ranked Stosur was making her first appearance at this U.S. Open warmup tournament after losing in the final last week at Toronto.
Sharapova will play the winner of the match between unseeded Daniela Hantuchova of Slovakia and No. 2-seeded Vera Zvonareva of Russia, the highest seed remaining in the tournament.
Fourth-seeded Andy Murray, the 2008 champion, advanced to the men's semifinals with a 6-3, 6-3 win over 10th-seeded Gilles Simon of France. The Scot, who has beaten Simon in seven consecutive matches, will play the winner of the match between second-seeded Rafael Nadal of Spain and No. 7-seeded Mardy Fish of the United States.
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