Paes and his Russian partner Elena Vesnina, seeded seventh, defeated the Taipei-Polish pair of Yung-Jan Chan and Mariusz Fyrstenberg 6-2 6-7 (7) 1-0 (4).
They will now take on the winner of the match between the Belarussian-Polish combine of Olga Govortsova and Marcin Matkowski and Natalie Grandin and Jean-Julien Rojer.
In men's doubles, Bopanna and his Pakistani partner Aisam-ul-Haq Qureshi got the better of the American-German team of James Cerretani and Philipp Marx 7-6 (3) 7-6 (4).
The fifth-seeded Indo-Pakistani pair will now be up against the winner of the match between Columbians Juan Sebastian Cabal and Robert Farah and the Australian-Belarussian combination of Paul Hanley and Dick Norman.
However, there was disappointment for Bopanna in the mixed doubles event. The Indian and his American partner Vania King, seeded fifth, lost 6-7 (1) 6-4 0-1 (7) to the Czech-German pair of Barbora Zahlavova Strycova and Philipp Petzschner.