The 14th seed will next face eighth-seeded American Madison Keys, who fought back from a 2-5 deficit in the first set to beat Russian Svetlana Kuznetsova 7-6(2) 6-2, in the last eight.
"I think I'm mentally tough right now. I know that my game is there so I know I just need to hit it like I can," Kvitova told reporters.
German Kerber twice failed to close out the match against Strycova before overcoming a late surge from the Czech to win 6-3 7-6(4). Kerber will face Elina Svitolina of Ukraine for a spot in the quarter-finals.
U.S. Open finalist Karolina Pliskova also secured a third-round spot after recovering from losing the first set against Daria Kasatkina of Russia to win 3-6 7-5 7-6 (7).
Pliskova, the fifth seed, will face Britain's Johanna Konta, who beat Hungary's Timea Babos 7-5 6-2 in a rain-affected match.
(Reporting by Nivedita Shankar in Bengaluru)
