Bopanna and his partner Pablo Cuevas of Uruguay defeated Thomaz Bellucci of Brazil and Maximo Gonzalez of Argentina 6-4, 7-6 (4) in the men's double category.
Sania and her Czech partner Barbora Strycova outplayed UK's Jocelyn Rae and Anna Smith 6-3, 6-1 in the women's doubles.
Bopanna and Cuevas will now take on Australians Alex Bolt and Bradley Mousley, who defeated Robin Haase and Florian Mayer 6-3, 5-7, 6-4.
The fourth seed Indo-Czech pair will next play the winners of the first round match between the Australian pair of Kimberly Birrell-Priscilla Hon and the Aussie-China team of Samantha Stosur and Shuai Zhang.
In the women's doubles, Sania and Barbora started on a positive note, winning in exactly one hour.
They did not face a single break point in the first set, winning 38 points to 32 to take the crucial lead.
In the second set, Rae-Smith broke the service of Sania and Barbora once but it was the latter who dominated after that, winning four out of seven break points.