The unseeded Indo-Uruguay pair pipped the Protea-American combination 6-7(6) 6-4 10-6 in the pre-quarterfinals.
They next face top seeded team of Henri Kontinen from Finland and Australia's John Peers, who ensured exit of Novak Djokovic and Victor Troicki with a 6-3 6-4 win.
The Monte Carlo Masters is fifth tournament Bopanna is playing with Cuevas and this is their best performance together after joining forces at the beginning of the season.
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