Gavrilova made a triumphant return to Grand Slam tennis, stunning 24th seed Dayana Yastremska in straight sets, 6-4 6-3.
The 26-year-old had been out of action for more than a year due to a debilitating foot injury, but wasted little time with the Ukrainian in what were rather chilly conditions.
Compatriot Sharma, meanwhile, had a tougher time of things against Russian Anna Blinkova, prevailing in three sets, 6-3 2-6 7-5.
Ajla Tomljanovic and Maddison Inglis were less fortunate in their first round matchups, going down in straight sets to Maria Sakkari and Ekaterina Alexandrova respectively.
On the men's side of the draw, Alex de Minaur and Jordan Thompson were both unable to replicate their US Open heroics.
Marco Cecchinato proved too much for de Minaur (7-6 6-4 6-0), while Radu Albot, too, eased through in straight sets (6-2 6-4 6-1).
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