The sixth seeded Indo-Romanian pair faced early resistance from the Slovakian-Chile combination before prevailing 7-5 6-2 in the second round contest that lasted one hour and 17 minutes.
Bopanna and Mergea saved eight of the nine breakpoints they faced in the match and broke their rivals four times.
Bopanna could face Leander Paes in the next match if the 43-year-old and his Polish partner Marcin Matkowski get the better of Henri Kontinen and John Peers.
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