Sharapova was broken in the first game but fought back to draw level before going on to take the first set with a hold to love.
The 30-year-old, with five grand-slam titles, made a quick start on Friday against the 196th-ranked Voegele, taking the first set and leading 4-0 in the second.
Voegele presented Sharapova with three match points when she made her eighth double fault when serving at 5-1 down and the Russian sealed the match with a forehand victor. Sharapova credited the victory to her aggressive game in the opening set, the momentum of which carried her all the way to the second set and the victory.
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Third seed and defending champion Peng Shuai coasted into the quarterfinals with a 6-2, 6-2 triumph over fellow Chinese countrywoman Wang Xiyu while American Christina McHale upset sixth-seed Greek Maria Sakkari, 6-0 6-4.
Sharapova took command at the end of the set.
Wildcard Sharapova advanced with five breaks of serve while losing her own once.
The Czech coughed up eight double faults as she lost 6-4 5-7 6-4 to world number 113 Zhu Lin of China.