The fourth-seeded pair of Sania and Elena disposed off a spirited challenge in the second set from unseeded Anna Chakvetadze of Russia and Melanie Oudin of the US to win 6-0, 7-6(4).
They now meet Czech Renata Voracova and Galina Voskoboeva of Kazakhastan, who beat England's Naomi Broady and Emily Webley-Smith 6-3,6-4.
Sania was doubtful starter for the doubles events after sustaining an injury on her left kneee which was aggravated during her first round singles loss on Tuesday.
Meanwhile, it was shocker for Indo-Pak Express Bopanna and Qureshi who went down to unseeded Colombians Juan Sebastian Cabal and Robert Farah 6-2, 2-6, 19-21 in a marathon three-setter which was curtailed due to rain delay.
The fourth-seeded Indo=Pak pair had won the Gerry Webber Open, in the lead-up to the Wimbledon.
Bopanna will be also playing in the mixed doubles alongside Sania. The two are seeded sixth and have received a bye in the first round.
Qureshi has paired up with Czech Kveta Peschke of Czech Republic and they are seeded fifth.