Cheteshwar Pujara led from the front in the series opener in Bridgetown, Barbados with half centuries in both the innings.
Chasing a modest 186, India were in doldrums at five for 68 before Pujara played a captain's knock to guide the visitors to a thrilling win.
Pujara, who struck a 50 in the first innings, was patience personified during his 96-run unbeaten knock in the second essay. He added 73 runs with number 10 batsman Shami Ahmed (27) to steer India after a spectacular collapse jeopardised the visitor's chances.
Apart from Pujara, Rohit Sharma seemed in good nick in Barbados with scores of 94 and 23, while Manoj Tiwary (40) and wicket-keeper Wriddhiman Saha (56) also made useful contributions in the first innings.
But going into the second match of the series, the major concern for India would be the performance of its top-order.
Both the openers -- Abhinav Mukund and Shikhar Dhawan -- have failed to provide India a good start, and to add to the woes Ajinkya Rahane looked a pale shadow of himself at the crucial number three position.
Pujara, however, took a giant stride towards Test breakthrough by virtue of his mature knocks and would be hoping to carry on his good form in the remaining matches.
Contrary to the inconsistent performance of their batting colleagues, the Indian bowlers, however, were right on the money in both the innings, dismissing the hosts for modest scores of 252 and 210.