The second-seeded Sharapova, who is chasing her second career Australian Open title at the age of 27, needed just 75 minutes to knock off the 21st-seeded Peng 6-3, 6-0.
Bouchard, seeded No. 7, had a tougher time against Irina-Camelia Begu but overcame a 61-minute second set to win 6-1, 5-7, 6-2.
The 20-year-old Canadian was runner-up at Wimbledon last year and has been the focus of some attention this week after a male interviewer asked her to "give us a twirl" following a second-round win.
The request drew wide criticism, including from tennis great Billie Jean King, who called it sexist.
Sharapova is 3-0 all-time against Bouchard, with each of their meetings coming in the past two seasons, including a three-set battle in the semifinals of the French Open last year.
"I felt like I was close with her. I remember that," Bouchard said, according to the WTA. "I remember I had chances and stuff. I just remember a really tough match. And I was pretty disappointed afterwards, so that's definitely motivating."