The fourth seed Indo-Romanian pair needed 70 minutes to get the better of the 2014 US Open champions 6-3, 7-6(10).
Bopanna-Mergea forced an early break to race to a 3-0 lead. They then held serve to take the first set easily.
However, the second set was neck-and-neck and, with no breaks of serve coming, needed the tie-breaker to decide it.
In the end, Bopanna-Mergea needed five match-points to seal it after Cilic-Moser had a set point.
They will take one the winner of the other first round match of the grass-court tournament between Andreas Seppi/Viktor Troicki and Mate Pavic/Michael Venus.
Image: India’s Rohan Bopanna and Florin Mergea of Romania in action at the Madrid Open on May 9, 2015.
Photograph: Clive Brunskill/Getty Images