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The third-ranked Russian, looking for her third title in Qatar, broke Stosur twice to lead 4-1 and then hit a backhand down the line to make it 5-1. She closed it out when the ninth-ranked Australian hit her return long.
Sharapova jumped out to 3-0 in the second set but Stosur staged a brief comeback. She had a chance to level at 4-4 but Sharapova hit five aces to win the game and won the match on a drop cord off her backhand.
"I was happy because Sam certainly stepped up her level in the second set, and despite, you know, losing a couple of more points than I would have liked to and losing my concentration, I really started playing well when I had to and being aggressive at the right times," she said.
"After not serving well for about a set and the beginning of the second, on the important points, when I had to, I came up with really great first serves. That helped me win."
Sharapova, who plays either Serena Williams or Petra Kvitova, has an outside chance of returning to No. 1 for the sixth time. But she has to hope Williams and top-ranked Azarenka falter.
If Williams reaches the semis, the 31-year-old American will be the oldest woman to reach No. 1. Chris Evert held the top ranking in 1985 just shy of her 31st birthday.
If Williams loses to Kvitova, Azarenka would retain the top spot by reaching the final. Sharapova would return to No. 1 if the other two falter and she wins the tournament. But the prospect of going No. 1 doesn't have the Russian in high anticipation.
"Is it great to have the opportunity to come back to that position? I mean, we'd be lying if we weren't. We're excited about having the chance," she said. "The ranking is always one of those things where it also depends on the other players' success and the amount of points and the tournaments that they play. That's out of my control, so that's the reason why, to me, maybe it's not as important as winning the match point of a Grand Slam. Whereas, you wake up on a Monday morning and the ranking system changed and you're No. 1 in the world."
Results:
Maria Sharapova (RUS x3) bt Samantha Stosur (AUS x8) 6-2, 6-4
Victoria Azarenka (BLRx1) bt Sara Errani (ITAx6) 6-2, 6-2
