Kohli struck 139 not out from 88 balls on Sunday while Shubman Gill made 110 as India chased down 296 with 50 balls to spare to take a 1-0 lead in the three-match series.
Rohit Sharma and Gill launched the chase with a 74-run opening stand before Sharma was dismissed in the 12th over.
Gill, the world's top-ranked ODI batter, then took control, maintaining the aggressive approach and bringing up his 10th ODI century with a sweep.
Kohli came in at first-wicket down and notched his 86th international century, while Gill struck 110 from 108 deliveries as the pair put on a 151-run stand off 132 balls that all but sealed the contest.
Roston Chase eventually removed Gill, but Ruturaj Gaikwad made 13 off 19 balls and helped Kohli steer India home without further alarm. Kohli sealed victory with successive boundaries.
Put in to bat, Justin Greaves and John Campbell gave West Indies a strong start, adding 135 from 117 balls for the opening wicket.
India struggled to find a breakthrough for much of the first 20 overs as the pair dominated the bowling and provided a solid platform.
Greaves, who was dropped by Sharma on 16, made India pay with a maiden ODI century and top-scored for the visitors.
The momentum shifted when Kuldeep Yadav dismissed Campbell in the 20th over, beating him in the air as he swept a catch to short fine leg. Yadav and Prasidh Krishna then applied the brakes as West Indies lost four wickets for 22 runs between the 31st and 36th overs.
Yadav finished with 4-40 on his ODI return but Amir Jangoo counter-attacked with 58 off 43 balls. With support from Chase, he helped West Indies recover from 198-5 and post 295.
