Williams has a 14-2 record and 13-match winning streak against Sharapova dating back to the 2005 Australian Open semifinals, beating the Russian most recently in last year's French Open final.
Any friendship the pair had at that stage soured when they traded personal barbs relating to their romantic relationships ahead of Wimbledon, where Sharapova made an early exit. Sharapova only played one match after that in 2013 and spent the latter months of the season recovering from a right shoulder problem.
Williams, meanwhile, had a spectacular year, winning 78 of her 82 matches and capturing 11 titles, including two majors.
As far as any grudges go, Williams said Thursday: "It's very difficult I think for anyone to be best buddies when you're so competitive."
Sharapova didn't back away from the comments she made about Williams at Wimbledon in a New York Times profile last month, giving the impression that the relationship between the pair was still cold. On Thursday, she said she'd used the newspaper interview to "clear the air."
Asked how she'd describe their rivalry now, Sharapova replied: "Well, I think I got to win a few times in order to call it rivalry."
"I haven't had a lot of success against her," she added. "It's the first tournament of the year. I came here wanting to play as many matches as I could and obviously wanting to play the best."
Australian Open champion Victoria Azarenka had difficulty closing out her quarterfinal match before beating Stefanie Voegele 6-4, 6-7 (7), 6-1.
The second-seeded Azarenka failed to convert eight match points in the second set three serving at 5-4 and another five in the tiebreaker before Voegele rallied to force a third. Azarenka raced to a 4-0 lead in the deciding set, dropped serve and then finally finished it off on her 10th match point.
"Yeah, I made it ... about 20 times harder than it should have been," Azarenka said. "I definitely should have closed it out a little earlier."
Azarenka will next face fourth-seeded Jelena Jankovic, who beat fifth-seeded Angelique Kerber of Germany 6-7 (8), 6-3, 6-1.
In men's second-round matches, Romanian qualifier Marius Copil beat third-seeded Gilles Simon 7-5, 6-3 and will next meet former No. 1-ranked Lleyton Hewitt, who ousted No. 6-seeded Feliciano Lopez 7-5, 6-3.
No. 8-seeded Jeremy Chardy beat Nicolas Mahut 7-5, 6-7 (4), 6-3 and fourth-seeded Kevin Anderson withdrew due to a stomach ailment before his scheduled second-round match against Australian wildcard entry Sam Groth.
In other Australasian tournaments, Milos Raonic and Eugenie Bouchard combined to give Canada a 3-0 win over Italy at the Hopman Cup in Perth, and Venus Williams and Ana Ivanovic progressed to the semifinals at the WTA Tour's ASB Classic in Auckland, New Zealand.
Canada (2-1) maintained a chance of qualifying for Saturday's Hopman Cup final from Group A when Raonic beat Andreas Seppi 6-2, 6-4 and Bouchard clinched it when Flavia Pennetta retired with a right wrist injury while trailing 4-0 in the first set.
Poland also won, beating Australia 2-1.
In Auckland, Venus Williams beat Garbine Muguruza of Spain 6-3, 6-3 and the second-seeded Ivanovic beat Kurumi Nara of Japan 6-2, 6-3.
Ivanovic will play her doubles partner, third-seeded Kirsten Flipkens of Belgium, in Friday's semifinals while Williams will play Jamie Hampton.
Flipkens beat Japanese qualifier Sachie Ishizu 6-4, 7-5 and the fifth-seeded Hampton beat Lauren Davis 6-2, 4-6, 6-4.