Sharapova, who was beaten by Simona Halep in the third round of the China Open last week, took a 4-3 lead in the first set before Linette hit back. But Sharapova broke her rival in the 12th game to take a 1-0 lead.
Ranked 15 places below world number 71 Linette because of her absence from the tour, the 30-year-old Russian appeared in control in the next set despite conceding an early break.
The five-times grand slam champion broke back twice and held on to set up a clash with Swiss qualifier Stefanie Vogele who beat Yulia Putintseva 6-2 6-3.
Meanwhile, defending champion Peng Shuai beat compatriot Wang Xiyu 6-2 6-2 while Sara Errani got past Brazilian Beatriz Haddad Maia 6-4 4-6 6-3.
(Reporting by Shrivathsa Sridhar in Bengaluru Editing by Jeremy Gaunt)
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