The five-time grand slam champion, soundly beaten by new world number one Simona Halep in the China Open last week, saw off Linette 7-5 6-3 to move into the last eight.
Sharapova has not added to her huge haul of titles since returning from a doping ban, but remains in the hunt for a 36th WTA Tour singles success after claiming the only break of the first set and two in the second to prevail.
The Russian's first-serve percentage was only 59 per cent, but she was able to overpower the Pole and saved nine break points to get the job done in two sets, setting up a meeting with Stefanie Voegele – who ousted Yulia Putintseva in the final match of the day.
Peng Shuai, a winner of singles and doubles titles at this tournament last year, cruised to a 6-2 6-2 win over Wang Xiyu in an all-Chinese second-round encounter.
Third seed Peng broke twice in each set to send her compatriot packing in an hour and 20 minutes. Peng will now face Sara Sorribes Tormo, who beat Wang Yafan 6-4 2-6 6-3.
Sixth seed Maria Sakkari crashed out, losing 6-0 6-4 to Christina Mchale, while Sara Errani, Aryna Sabalenka and Zhu Lin all advanced.