Cibulkova, the highest seed remaining, defeated Yuliya Beygelzimer 3-6, 6-0, 6-0. She won about two-thirds of the points in the final two sets.
She'll next go against Sorana Cirstea, who got by Olga Govortsova 2-6, 6-1, 6-3. Cirstea claimed 52 percent of the points off Govortsova's serve and finished with six service breaks.
Seventh-seeded Sara Errani ousted No. 2-seeed Julia Goerges 6-2, 6-3 by winning 60 percent of the points over the course of the match.
Carla Suarez Navarro was a 6-4, 6-1 winner over Simona Halep in the other quarterfinal. Suarez Navarro had eight breaks and won an incredible 70 percent of the points off Halep's serve.
Cibulkova won the only previous time she's played Cirstea while Suarez Navarro is 2-1 against Errani, although they haven't played since 2008.