Bhupathi, who is playing with Luxembourg's Gilles Muller, beat the Australian team of Alex Bolt and Andrew Whittington 7-6 (4) 3-6 6-4 in his tournament-opener at court seven.
In the two hour and 13 minute contest, Bhupathi and Muller managed to save 12 breakpoints out of 15 they faced.
It was Bhupathi's third competitive event since bowing out of 2015 Wimbledon championships. He had missed the entire second half of the last year as he was busy with the conduct of his ambitious International Premier Tennis League (IPTL).
The 42-year-old Paes made an exit with French partner Jeremy Charady after losing his first round 3-6 4-6 to Colombian 12th seeds Juan Sebastian Cabal and Robert Farah.
The Australians got four set points with a forehand
volley winner and then pocketed the set on the first, courtesy a forehand unforced error from Bopanna and Cuevas.
With the match level at one set apiece, the decider went on serve in the first six games before Bolt and Mousley broke Bopanna's serve with a forehand winner to take a 4-3 lead.
Bopanna later blamed a "horrendous call" at a crucial juncture in the match from the chair umpire as the reason behind his early ouster from the men's doubles event.
The contentious call was made on Cuevas' serve, which was broken in the seventh game of the final set. On break point, a volley from Bolt sailed wide but the chair umpire ruled that the ball had brushed Bopanna's racket on the way out.
Protesting the call, Bopanna and Cuevas were involved in an animated conversation with the chair umpire at the change of ends.
He later tweeted: "Very hard to digest a loss in a match when the REF decides the outcome.Reactions of opponents says it all.@AustralianOpen #MostShockingCall.
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