Stosur, the No. 1 seed, escaped Kristina Mladenovic 6-4, 2-6, 7-6 (9-7). Stosur dropped serve twice in both of the first two sets but was able to convert her only break chance of the third set before deciding the match in the extended tiebreaker.
Flipkens, seeded second, edged by Storm Sanders 4-6, 7-6 (7-3), 7-6 (7-5) despite three breaks in each of the first two sets and two more in the third.
Both of the matches took more than 2 1/2 hours to complete.
Wednesday's other two matches were stopped early due to retirements. Estrella Cabeza Candela was ahead 4-6, 6-2, 4-1 when third-seeded Elena Vesnina was unable to continue and Garbine Muguruza built a 4-1 lead before Yanina Wickmayer was forced to stop play.
Celebrity Families of 2014 [PHOTOS]