Fourth seeds Paes and Stepanek won the match, stretched to second day after being suspended due to rain, with a 7-6 (6) 6-4, 6-7 (1), 6-4 win after battling hard with the Italian-Israeli pair for three hours and five minutes.
The Indo-Czech pair had secured the first set yesterday before skies opened up, forcing suspension of play.
They next face local guy Jamie Delgado and Australian Matthew Ebden.
The Indo-Slovakian pair lost 7-6 (2) 4-6 4-6 to Mark Knowles of Bahamas and Sabine Lisicki of Germany.
Bhupathi's challenge in men's doubles is still alive as he has moved to the second round with Autrian partner Julian Knowle.
They are up against Amercian Nicholas Monroe and Simon Stadler of Germany.