The 96th-ranked Zarina Diyas of Kazakhstan upset Wimbledon semifinalist Flipkens 6-2, 6-7 (3), 6-3, while Casey Dellacqua of Australia stunned Vesnina 7-5, 3-6, 6-3. Dellacqua broke serve seven times as Vesnina made nine double-faults.
American trio Madison Keys, Vania King and Alison Riske moved into the second round.
The eighth-seeded Riske beat Teliana Pereira of Brazil 6-3, 6-3, and Keys won the last four games to defeat Maria-Teresa Torro-Flor of Spain 7-6 (2), 6-2.
King outlasted Paula Ormaechea of Argentina 5-7, 6-2, 6-1. A semifinalist in Bogota last month, King took a 4-0 lead in the second set and won the last five games in the decider.
French wild-card entry Claire Feuerstein and Ajla Tomljanovic of Croatia also won their first-round matches.
Feuerstein saved 10 of 12 break points in a 6-2, 6-4 win over Sharon Fichman of Canada, while Tomljanovic hit six aces to dismiss Kristina Mladenovic of France 6-4, 6-0.