Beaten in a five-setter in his previous meeting with Nishikori in last year's U.S. Open, twice winner Cilic drew first blood but Nishikori clawed his way back to seal a 4-6 7-6 6-2 win in more than two-and-a-half hours.
Nishikori next meets the winner of Tuesday's match between Colombia's Alejandro Falla and Portuguese Rui Machado.
Top seed Tomas Berdych begins his campaign on Tuesday against Portugal's Frederico Gil.
Standing a good eight inches taller than an opponent ranked 84 places behind him, world number 14 Cilic could not press home his advantage after winning the first set.
Nishikori made a remarkable comeback in the second set which went to a tiebreaker he won without dropping a point and the Japanese broke Cilic twice in the decider.
Germany's Bjorn Phau, Frenchman Stephane Robert and American Robert Kendrick also reached the second round.
(Reporting by Amlan Chakraborty in New Delhi; Editing by Ed Osmond)