The fourth-seeded pair of Sania and Elena overcame a second set resistance from unseeded Anna Chakvetadze of Russia and Melanie Oudin of the US to win 6-0, 7-6(4).
They now play Czech Renata Voracova and Galina Voskoboeva of Kazakhastan, who defeated England's Naomi Broady and Emily Webley-Smith 6-3,6-4.
Sania was doubtful for the doubles events owing to a painful left knee injury she aggravated during her first round singles loss Tuesday. She was to decide on her participation after an ultrasound scan.
Meanwhile, the 2010 US Open runners-up Bopanna and Qureshi went down to unseeded Colombians Juan Sebastian Cabal and Robert Farah 6-2, 2-6, 19-21 in a marathon match which was cut down to three sets due to rain delay.
The fourth-seeded India-Pakistan pair had won the Gerry Webber Open, their second ATP title together, in the tune-up to the Wimbledon.
Bopanna is still in contention in the mixed doubles where he is partnering 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.
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