Ranked 52, Giorgi made a sluggish start as Siniakova won the opening set 6-3.
The signs looked ominous as the Czech pushed for victory in the second set, only for the Italian to find her rhythm and win it 8-6 on a tie-break.
Giorgi then found another gear in the deciding set, winning 6-2 to set up a match round clash against Ekaterina Makarova of Russia.
Jonathan Erlich reached the third round of the Men’s Doubles when he and partner Marcin Matkowski stunned second seeds Kubot and Melo.
Erlich of Israel and Matkowski lost the first set on a tie-break before levelling the match after winning the second 6-4.
They maintained their momentum to win the next two on tie-breaks to progress.
Diego Schwartzman, the 14th seed in the Men’s Singles, tumbled out in the second round after going down in straight sets against Jiri Vesely.
