Vinci struggled early in the match but ultimately prevailed over Germany's Julia Goerges, 2-6, 6-4, 6-3 at Rexall Center, advancing to the second round.
Joining her there was Stephens, who took out Frenchwoman Kristina Mladenovic 6-2, 7-6 (7-5).
Dominika Cibulkova, coming off a victory at Stanford last month, moved on as well, besting fellow Slovakian Jana Cepelova 7-5, 6-4.
Czech Klara Zakopalova, Russian Anastasia Sergeyevna, Dutch qualifier Kiki Bertens and American Varvara Lepchenko also each notched a victory Monday.
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