The unseeded Ind-Uruguay pair fought past the rivals 6-3, 6-7(7), 10-2 in an hour and 27 minutes.
They had played in two Masters series event -- Indian Wells and Miami -- and lost in the first rounds in both.
Bopanna and Cuevas next face fifth seeds Raven Klaasen from South African and American Rajeev Ram who got a bye in the first round.
Bopanna has made two finals this season -- winning the season-opener Chennai Open with Jeevan Nedunchezhiyan and ending runner-up with Matkowski at Dubai Championships.
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