Paes and his Czech partner Radek Stepanek, who are seeded fourth, cruised into the third round with a 6-4, 6-4, 6-3 win over Britain's Jamie Delgado and Australia's Matthew Ebden.
Bhupathi and his Austrian partner Julian Knowles, seeded eight, came back well from a set down to beat American Nicholas Monroe and German Simon Stadler 3-6, 6-4, 6-4, 6-2 in a battle that lasted two hours and 22 minutes.
Bhupathi and Knowles will next face Canadian pair of Jesse Levine and Vasek Pospisil. For Bopanna and his French partner Edouard Roger-Vasselin it was a tough fight against the German-Czech pair of Daniel Brands and Lukas Rosol in a five setter that lasted over three hours.
Bopanna and Roger-Vasselin beat Brands and Rosol 6-3, 5-7, 7-6 (7-4), 6-7 (8-10), 6-4 to set up a third round clash against the third seed pair of Austria's Alexander Peya and Bruno Soares from Brazil.