Rain reduced the third match of the series in Thiruvananthapuram to a 48 ball lottery and Kohli's side were stronger in the boundary chase and smarter in the bowling battle.
While Manish Pandey hit 17 off 11 balls, Hardik Pandya made 14 and Kohli 13 in India's 67-5, the home side owe a particular debt to man-of-the-match bowler Jasprit Bumrah in the six run victory.
His two overs cost just nine runs while he claimed the key wickets of big-hitting Colin Munro and Henry Nicholls. Indian coach Ravi Shastri hailed Bumrah, who cannot get a Test place, as a "thinking bowler".
Pandey and Kohli both hit a six and a four in quickfire innings that New Zealand just could not match as they reached 61-6.
Pandey went to a spectacular catch off Trent Boult that highlighted a night of brilliant fielding.
He fired the ball toward the long-on boundary producing a full length dive by Mitchell Santner who deflected the ball to stop it going over the line and set up the catch by Colin de Grandhomme.
New Zealand started aggressively. Munro hit his first ball for a six, but made only seven before he was caught in the second over trying to hit Bumrah.
Captain Kane Williamson and Glenn Phillips went in successive balls in the fifth over leaving New Zealand facing a struggle to defend their number one place in 20 over cricket.
Though the result could have gone either way with just two balls remaining, New Zealand reached just 61-6 with de Grandhomme making an unbeaten 17.
"You would be lying if you said you were not tense," said Shastri. "They saved the best for the last in terms of fielding, the electricity was there."
Shastri said that while 67 was not an automatic match-winning score, the fielding and good start by the bowlers had put pressure on New Zealand.
India had never beaten New Zealand in a T20 match going into this series. New Zealand's defeat means Pakistan are the number one ranked 20 over nation again. New Zealand also lost a one day series in India but Williamson said they had still come out of the tour "stronger".
"It is a difficult place to play," said Williamson. "They are possibly the best team in the world."
By many counts, India are the best team. They lead the Test rankings, while Kohli is top-rated batsman in one-day and T20 internationals. Bumrah is the number one bowler in T20.
The Indians have not lost a home series in any format since being beaten by South Africa in one day games in 2015.
"We have won a few on the trot and it makes us very proud," said Kohli.
India must now prepare for a three Test series against Sri Lanka starting November 16 in Kolkata.