Davis, a qualifier, saved all three break points she faced in a 6-1, 6-3 win over Cirstea. She established control with three first-set breaks and sealed the win with the only break of the second set.
She goes into a quarterfinal match against eighth-seeded Sloane Stephens, who advanced on a 6-4, 6-0 win over Simona Halep with three breaks in each set. Stephens lost only 10 points in the second set.
Elena Vesnina defeated fourth-seeded Yaroslava Shvedova 4-6, 6-2, 6-1 in another second-round match. Vesnina saw just three breaks points -- saving them all -- over the second and third sets.
Ninth-seeded Mona Barthel got by Chanelle Scheepers 7-6 (7-3), 7-5 in Tuesday's other second-round match.
Celebrity Families of 2014 [PHOTOS]