Williams, the American former world No. 1, was originally set to play Monday but her match was washed out.
She wasted little time during the make-up, however, dispatching Croat Mirjana Lucic-Baroni 6-3, 6-3 in 79 minutes.
Belgium's Flipkens, meanwhile, was upended by Russian Olga Puchkova, who reached the second round with a 2-6, 6-4, 6-4 victory at the Federacao Catarinense de Tenis.
Advancing to the quarterfinals was Slovak Jana Cepelova, who surprised eighth-seeded Annika Beck of Germany 7-6 (8-6), 6-4 in Tuesday's the lone second-round match.
Other opening-round winners included fifth-seeded Slovak Magdalena Rybarikova, seventh-seeded Frenchwoman Kristina Mladenovic, German Tatjana Malek, Brazilian wild card Beatriz Haddad Maia, Hungarian Timea Babos, and Spaniard Garbine Muguruza.