The 318-run victory in the series-opener was India's biggest-ever away win by runs in test cricket.
It was Kohli's 27th win from 47 tests as India captain, equaling MS Dhoni's record for wins at the helm of the national team. Dhoni led India to 27 wins in 60 tests as captain.
"It's a blessing that I am in a position to contribute to my team in more than one way," Kohli said. "I am taking the decisions but it is up to the individuals to do the execution."
West Indies fell far short of the highly improbable winning target of 419 but managed a mini-recovery in its second innings after slumping to 15-5. Paceman Bumrah bowled eight overs, including four maidens, to do most of the damage for India, hitting the stumps to claim four of his wickets.
Earlier, India resumed Day 4 on 185-3 with Kohli and Ajinkya Rahane already having secured half-centuries. India declared its second innings at 343-7 with Rahane out for 102, his 10th test century. Kohli failed to add to his overnight 51.
"We wanted to just bat, bat and bat," Rahane, voted player of the match, said of his partnership with Kohli.
Hanuma Vihari scored 93 before he was caught behind off Jason Holder, ending a 135-run partnership with Rahane for the fifth wicket. Roston Chase took 4-132.
Holder had no excuses for his team's performance.
"We players need to take more responsibility ... we fell short by quite a few runs," Holder said. "Bumrah bowled some exceptional deliveries, he's a very good bowler, but we have to find ways to negate it."
"There's no point dropping our heads — that's not the solution. We have to find a way to bring ourselves back out of this defeat, try to rally around one another and try to come back a lot better in Jamaica."
India scored 297 in its first innings before dismissing West Indies for 222.
The second and final test starts Friday at Kingston, Jamaica.
"We had to come back into the game three or four times. We know West Indies will come hard at us," Kohli said, "and we will try to plug the gaps from this test."