World number 33 Gavrilova next faces Czech Barbora Strycova, who beat Alize Cornet 6-3 6-2.
Spaniard Garbine Muguruza, seeded 14, beat her Wimbledon conqueror Alison Van Uytvanck 6-4 6-0 to set up a second-round clash with Swiss qualifier Viktorija Golubic.
"I remember the match from Wimbledon, like a very tough match for me... It's always good to play again the same person that beat you... feeling good you did something much better than the previous match," Muguruza said.
Briton Johanna Konta's patchy form continued as she fell 7-5 6-4 to last year's finalist Ashleigh Barty.
American Coco Vandeweghe was forced to retire from her match with Donna Vekic after injuring her right ankle while trailing 6-4 3-6 2-1. Croatia's Vekic faces Anett Kontaveit next.
World number one Simona Halep, scheduled play Dominika Cibulkova on Tuesday, faces a struggle to be fit after suffering a back problem while training on Monday.
"I don't know why it happened. It's been a difficult 24 hours, but I hope to be ready tomorrow," Halep said.
(Reporting by Shrivathsa Sridhar in Bengaluru; editing by John Stonestreet)
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