Bopanna and Canada's Nestor, seeded seventh, fought off a spirited challenge from Marcos Baghdatis and Marinko Matosevic 7-6 (2) 7-5 in their opening round in one hour and 46 minutes.
There was not much to differentiate between the two teams as they hit an equal number of aces (10) and committed same number of double faults (5). Bopanna and Nestor won 87 points, only eight more than their rivals.
Earlier, Bhupathi and Austrian partner Melzer suffered a 4-6 3-6 defeat against Argentine team of Diego Schwartzman and Horacio Zeballos in only 67 minutes.
Bhupathi continues to struggle in his comeback as he returned to on court action after a nine-month break during which he set up IPTL.
He played his first tournament at Chennai Open where he had lost in the second round with compatriot Saketh Myneni.
The first set was a close affair with both team holding serve till the ninth game. Bhupathi was terrific at the net, specially with his wristy drop volleys.
The 6 feet 2 inch Zeballos was serving bombs and his service made a big difference to the outcome of the match.
On Melzer's serve, the Austrian and Bhupathi saved the first two but Melzer sent a forehand to net on the third chance and threw his raquet in frustration.
