Bumrah claimed three wickets and effected a crucial run out of dangerman Tom Latham in the 48th over to make the visitors falter in their 338-run chase at Kanpur's Green Park Stadium.
The Kiwis, who looked like favourites with 35 runs required from the last 24 balls, ended on 331-7 despite Latham's 52-ball 65.
"It was a pretty close game. Credit to New Zealand, they fought hard in this series and forced us to play our best cricket," Kohli said at the post-match presentation.
"I decided that I let the bowler do what he wants to do, and that's why I was calm...glad the bowlers came out on top," Kohli, who led India to their seventh successive bilateral series triumph, added.
Earlier sparkling centuries by opening batsman Rohit Sharma and Kohli, who made 113, steered India to 337-6 after being put in to bat.
Sharma, who smashed 147 off 138 balls, and Kohli became the first pair in ODIs to record four double-century stands between them after putting together 230 runs for the second wicket.
"The aim is always to help the team win games, regardless of how many runs I score. Very happy we could get another one in the bag," said Kohli.
But it was the superb death bowling show by Bumrah that stood out in the big-scoring game. Leg-spinner Yuzvendra Chahal also chipped in with two crucial wickets.
Kiwi opener Colin Munro hit a quickfire 75 to give the visitors a blazing start despite losing his partner Martin Guptill for 10 to Bumrah.
Munro put together a crucial 109-run partnership for the second wicket with skipper Kane Williamson, who made 64, as the pair took the attack to the opposition bowlers.
The left-handed Munro, who smashed 8 fours and 3 sixes in his 62-ball knock, took a special liking to Indian pace spearhead Bhuvneshwar Kumar, whose first five overs cost 51 runs.
Chahal soon spun into action to send Munro and Williamson trudging back to the pavilion as New Zealand stumbled in their chase.
It was then that the left-handed Latham got down to the grind, forming crucial partnerships with Ross Taylor and Henry Nicholls and take the attack to the opposition.
Kumar bowled Nicholls for 37 and Bumrah soon took over to dash New Zealand's hopes of winning their first series on Indian soil.
New Zealand bowlers also made their presence felt after the marathon stand between Sharma and Kohli ended in the 42nd over. Left-arm spinner Mitchell Santner got Sharma's prized scalp.
Paceman Tim Southee sent Kohli packing but not after the in-form batsman, who became the quickest to get to 9000 runs during his 106-ball knock, registered his 32nd ODI century.
Southee, fellow quick Adam Milne and Santner took two wickets each.
"Some really good cricket played and great spirit. Credit to Virat and team, they were the better team, just. Lots of promising signs for us," said Williamson.
"Tom Latham batted with such clarity in his mind, it was great to watch. Also Henry Nicholls. Ross is not new but he's playing nicely as well. Bowlers stuck it out as well," he added.
The action now shifts to the three Twenty20 internationals starting November 1 in New Delhi.