Sania and her American partner Bethanie Mattek-Sands overcame a fighting pair of Jarmila Gajdosova and Sabine Lisicki 6-1 3-6 10-7 in the second round.
They will now fight it out with Sara Errani and Roberta Vinci for a place in the semifinals.
In the men's doubles second round, unseeded Bopanna and Ram were knocked out by formidable second-seeded pair of Marcel Granollers and Marc Lopez 6-4 7-6(5).
Somdev Devvarman was floored by world number one Novak Djokovic in the men's singles while Leander Paes and Mahesh Bhupathi had exited from the doubles after suffering second-round defeats with their respective partners.