Davis, an 18-year-old American qualifier, stunned second-seeded Yanina Wickmayer 6-1, 6-1 to reach the quarterfinals.
Flipkens, playing in the last of the first-round matches, eliminated No. 1 seed Dominika Cibulkova 2-6, 6-3, 7-5.
Mona Barthel, who at No. 3 is the highest seed remaining in the event, earned a quarterfinal berth with a 6-2, 6-1 decision over Petra Rampre. Another quarterfinal spot went to Kristina Mladenovic, who rallied to defeat Annika Beck 4-6, 6-4, 6-3.
Celebrity Families of 2014 [PHOTOS]