A target of 337 left the hosts facing a daunting task, and they ran out of steam despite veteran Virat Kohli's battling knock of 124.
India started shakily and lost wickets at frequent intervals before Kohli gave home fans hope as he combined with Nitish Kumar Reddy (53) in an 88-run partnership.
Kristian Clarke removed Reddy to stymie India's momentum, and the required run rate quickly climbed, but Kohli found another able partner in Harshit Rana as the pair stepped on the gas.
Kohli reached a record-extending 54th ODI century, and Rana hit a whirlwind 52 off 43 balls, before a decisive over from Zak Foulkes broke India's resistance.
Foulkes tempted Rana with a full toss, which was caught by Henry Nicholls at long-on, before an edge from Mohammed Siraj carried through to wicketkeeper Mitchell Hay on the next delivery.
The final blow was dealt two overs later, when Kohli miscued a shot and sent the ball sailing into the hands of Mitchell at long-off.
"After the first game, coming here 1-1, the way we played was disappointing. A lot of areas need improving. Virat's batting and Harshit's batting have been positives," India captain Shubman Gill said.
Mitchell and Phillips smash tons
New Zealand were struggling at 58-3 after being put into bat on a Holkar Stadium wicket known for producing high-scoring games.
However, Mitchell took the attack to the hosts' bowlers and was well supported by Phillips as they added 219 off 188 balls.
Mitchell, who made 84 in the first ODI and 131 in the second, hit 15 fours and three sixes to post a career-high ODI score of 137 and record his fourth century in India.
"It's really nice to contribute and to win over here. The way we built partnerships, with (Will) Young and GP (Phillips)," said Mitchell, who was named Player of the Series.
"Always nice to get stuck into moments with your mates. I'm just trying to concentrate on being present, watch the ball and repeat, and make some good decisions. GP is a mate, awesome to see him get a 100."
New Zealand had lost their seven previous ODI series in India.
"Always a pressure to come and play here. First time a NZ team has won an ODI series here, that's special," New Zealand captain Michael Bracewell said.
"You always hope to come here and play good cricket. As a group, we've stuck to what we do well. We're the sum of all the parts of the team and work as a group. We're a small country from the bottom of the world and take on big ones."
The teams next meet in a five-match Twenty20 International series from January 21st to 31st, a key part of their preparations for the T20 World Cup, which begins on February 7th and will be co-hosted by India and Sri Lanka.