All-rounder Krunal Pandya made a memorable debut, claiming the important wicket of Kieron Pollard before returning to score 21 not out off just nine balls, including the winning runs.
Despite the presence of a host of swashbuckling batsmen on both sides, the bowler-dominated contest featured only two sixes - one from each team.
Put into bat, the first 49 runs cost West Indies the top half of their batting order at the midway stage of their innings.
Opener Shai Hope fell victim to a comical run out when he and Shimron Hetmyer both ended up being on the non-striker's end.
Hetmyer outsprinted Hope to secure his wicket but could not make the most of it and fell for 10.
Pollard hit the only six of the innings but Pandya sent him back before the batsman could inflict more damage on the Indians.
Kuldeep Yadav then ripped the heart out of the West Indies middle order with his triple strikes as the touring batsmen struggled to read the unorthodox left-arm spinner.
Fabian Allen topscored for the tourists with 27 to take them past the 100-mark.
India's reply was equally wobbly with the hosts looking ill at ease against the pace of Oshane Thomas.
Bowling consistently above 145 kph, the paceman drew first blood when he dismissed India captain Rohit Sharma caught behind for six. West Indies successfully reviewed the original not-out decision after the on-field umpire had failed to detect the slight edge.
Thomas devoured Shikhar Dhawan, the other opener, with another thunderbolt and then skipper Carlos Brathwaite struck twice to jolt the hosts.
Karthik first revived India's chase with Manish Pandey (19) and then found an able ally in Pandya, whose clever and nerveless batting stood out.
The teams move to Lucknow for the second match on Tuesday.
(Reporting by Amlan Chakraborty in New Delhi; editing by Pritha Sarkar.)
