West Indies reached 315-5 in the final one-day international after being put in to bat but the decent total proved insufficient as India finished at 316-6, winning with eight balls to spare.
"We just needed a small partnership in the end and knew the opposition would crumble," Kohli said. "It is good to see Jadeja finishing the game. It is about identifying how players react to pressure. We have a vision regarding the (Twenty20) World Cup (in 2020) and we will keep chasing it."
In a match featuring several key partnerships, Kohli put on 58 runs with Jadeja for the sixth wicket after India was reduced to 228-5.
Lokesh Rahul (77) shared 122 with fellow opener Rohit Sharma (63) to give India a good start.
West Indies' Nicholas Pooran (89 runs in 64 balls) and Kieron Pollard shared a 135-run stand for the fifth wicket. Pollard smashed seven sixes and was 74 not out in 51 balls.
Jason Holder (1-63) and Alzarri Joseph (1-53) delivered the double breakthrough Sunday against India. Sharma was caught behind off Holder after a second consecutive 100-plus stand with Rahul.
Rahul reached his half-century off 49 balls and put on 45 runs with Kohli before he was out pulling a short delivery off Joseph.
That triggered a middle-order collapse as India lost Shreyas Iyer (7), Rishabh Pant (7) and Kedar Jadhav (9).
Iyer was caught in the deep after a half-hearted pull against Keemo Paul (3-59), who also accounted for Pant — bowled off a thick inside edge.
But Kohli motored along and reached his 55th ODI half-century off 51 balls and, with Jadeja, took India within sniffing distance of victory.
Kohli was bowled by Paul — with another inside edge — but Jadeja held his nerve to see India across the finish line. He was helped by Shardul Thakur who hit 17 not out off six balls, including two fours and a six.
Earlier, Pooran and Pollard's 135-run stand was the highest fifth-wicket partnership for West Indies versus India in ODIs, going past 109 by Marlon Samuels and Ricardo Powell in 2002.
"There is a lot of exciting talent in our team and we have seen what they are capable of," Pollard said.
Put into bat, West Indies got off to a steady start. Openers Evin Lewis (21) and Shai Hope (42) put on 57 runs before both fell in the space of 13 runs.
Shimron Hetmyer scored 37 off 33, including two fours and two sixes, to give the West Indies some momentum.
Roston Chase (38) was promoted to No. 3 and he put on 62 runs with Hetmyer for the third wicket. The tide again turned in India's favor as debutant pacer Navdeep Saini (2-58) struck two blows to leave West Indies reeling.
Pooran and Pollard came together at 144-4 and started comparatively slowly, putting on 50 off 46 balls. They accelerated from 197-4 at the start of the 40th to 279-5 in the 48th over, when Pooran was eventually caught in the deep.
Pooran hit 10 fours and three sixes to bring up his second consecutive half-century. Pollard's half-century came off 44 balls.
West Indies scored 77 runs in the last five overs.
West Indies won the first ODI in Chennai, while India fought back with a win in Visakhapatnam