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India's seamers Bhuvneshwar Kumar and Jasprit Bumrah conceded just 11 off the first two overs and had picked up two wickets by then, including in-form opener Colin Munro for just seven. That put New Zealand behind the eight-ball from early in the chase.
New Zealand's batsmen struggled to make much headway. Captain Kane Williamson was run out by a direct hit from Hardik Pandya running to the non-strikers end.
Glenn Phillips briefly threatened in getting to 11 before holing out to deep mid wicket, while Colin de Grandhomme, after one flat six which whistled over the boundary, finished on 17 off 10 balls.
New Zealand needed 32 off the last three overs and 19 off the last bowled by Pandya.
De Grandhomme's second six of the innings, lifting Pandya over mid wicket, off the third ball left New Zealand needing 12 off the last threes, but the bowler held his nerve.
Batting was a battle for India as they fought their way to 67 for five.
They managed just four fours and three sixes in their eight overs, the most notable contributions coming from captain Virat Kohli (13 off six balls), Manish Pandey (17 off 11) and Pandya (14 not out off 10).
The pitch was tacky and green, spinner Mitchell Santner's first ball gripping, turning and taking a chunk of turf out of the pitch.
Trent Boult found swing and bowled impressively; Tim Southee made heavy use of his slower ball and nabbed two wickets, but legspinner Ish Sodhi struggled with his control, and cost 23 while picking up two wickets.
The ball seemed to follow Santner in the field.
He took the first two catches off consecutive Southee deliveries in the second over, a skier to point followed by a fine low grab running in at deep square leg to remove Rohit Sharma.
But Santner's most spectacular piece of work came at long on when he dived to his right to knock up a ball to team mate de Grandhomme to dismiss Pandey in the final over.
The 50,000 crowd in a flash new stadium deserved to see some cricket, even if eight overs a side is a hit or miss event. But it turned out to be a mini-thriller.
"It was quite hard to know what a good first innings score was on that surface but credit to the ground staff for getting the game ready," Williamson said.
"It was a tricky surface and runs on the board on that surface were handy.
"Credit to India, they deserved to win the series, although it came down to the last couple of deliveries. It was just a shame we didn't get across the line in this decider."
Williamson liked the fight his team had shown but pointed out there were good lessons to be learnt by his players.
"You have to be at the top of your game to win a series over here. One-off games yes that's great, but you do need to get better throughout.
"That's why this team (India) is possibly the best team in the world because they challenge you in all areas and put you under a lot of pressure."
Classy seamer Bumrah was named man of the match, and the series, during which his economy rate was a fine 6.9 an over.
"It was a good day, very happy," he said.
"We wanted to stick to our basic plan. After watching the first innings we saw they were using their slower bowlers a lot.
"Our slower bowling wasn't gripping as much so we had to stick to backup plans and use the yorker and mixed our deliveries in the middle."
Kohli said his team knew New Zealand would scrap hard.
"Ever since the one-dayers started we've expected them to give us a good fight," he said.
"We were nervous whether we would be able to get enough runs on the board. 67 turned out to be a good total, plus we saved a few in the field."
Kohli bowled key man Bumrah in the seventh over, not saving him for the last as had been anticipated.
That proved a sound move, Bumrah finishing with two for nine off his two overs. That meant 19 were needed off the last, which was beyond New Zealand.
Kohli said after India lost their first two wickets quickly they rejigged their target to 65.
"It was by no means a match-winning score but if we fielded well, had a couple of good overs at the start then the pressure is on the opposition."
New Zealand won praise from India's coach, former international Ravi Shastri.
"This was a good win because it was against a very good side," he said.
"This side (New Zealand) can go places. New Zealand play well as a unit and competed excellently."
New Zealand's home international season starts on December 1, with the opening day of the first of two tests against the West Indies.
Pakistan come in the New Year for a limited-overs tour, New Zealand play a T20 tri-series against England and Australia before England provide the main course of the summer, two tests and a batch of ODIs.
