In a win that sets up a second-round match against Russian No. 1 Daria Kasatkina, Sharapova brushed aside her Australian opponent Gavrilova in the first set. Gavrilova raced to 4-2 in the second set, but Sharapova won four straight games and closed out the match.
Sharapova has rarely played the Russian tour events during her career, and when she has she's never advanced past the quarterfinals.
Sharapova won her only previous meeting with Kasatkina in straight sets in Montreal in August.
The 34-year-old Zvonareva returned to the top 100 in the rankings this month after a six-year absence.