Ranked 78, the Italian secured a 6-4, 6-3 victory over the Russian qualifier and will face Australia Ashleigh Barty in the next round.
Giorgi reached the final of the Sydney International Open as a qualifier last week, winning six matches without dropping a set.
The Federation Cup player will be determined to better her best performance at the Australian Open which saw her reach round three in 2015.
Wednesday sees Diego Schwartzman of Argentina return to action in the Men's Singles. He takes on Casper Ruud of Norway with a place in round three awaiting the winner.