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.