Babos survived 7-5, 3-6, 6-2 to set up a quarterfinal duel with Monica Niculescu in what will be a meeting of unseeded players.
Niculescu defeated Yanina Wickmayer 6-4, 3-6, 6-1 to earn her quarterfinal berth.
No. 5 seed Anastasia Pavlyuchenkova moved closer to a possible fourth career title by cruising past Tereza Mrdeza 6-1, 6-1.
The other two second-round matches saw No. 8 seed Ayumi Morita fight off Galina Voskoboeva 6-4, 7-6 (7-3) and seventh-seeded Urszula Radwanska take out Donna Vekic 6-4, 6-1.
Notable deaths of 2014 [PHOTOS]