Seventh seeds Paes and Hingis subdued the French pair of Edouard Roger-Vasselin and Alize Cornet 6-4 6-2 in the second round match, late on Saturday.
They are now up against Artem Sitak of New Zealand and Anastasia Rodionova of Australia.
Bopanna and his Spanish partner suffered a tame 2-6 4-6 defeat against sixth seeded pair of Horia Tecau from Romania and Slovenia's Katarina Srebotnik in their second round.
Mahesh Bhupathi was eliminated earlier from the event when he lost his first round with Russian partner Alla Kudryavtseva.
On Monday, three Indian players will try to make the quarterfinal grade.
Top seeds Sania and Martina Hingis will take on 16th seeds Spanish team of Anabel Medina Garrigues and Arantxa Parra Santonja in the women's doubles third round.
In the men's doubles, 11th seeded Paes and Daniel Nestor are up against Austria's Alexander Peya and Brazilian Bruno Soares in their third round.
Bopanna and his Romanian partner Florin Mergea, seeded ninth, face Poland's Lukasz Kubot and Belarussian Max Mirnyi.
In the boys singles, Sumit Nagal will open his campaign against Juan Pablo Ficovich from Argentina.
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