Chasing 161 for victory, India were 75-4 after nine overs when rain forced the players off the ground at McLean Park.
Even after the rain relented, play could not resume because of the wet outfield, and it turned out to be the par score under the Duckworth-Lewis method governing such weather-hit contest.
"We would have liked to win this game, but that is how it is," India captain Hardik Pandya said.
"We don't mind getting a trophy and going back with a win."
New Zealand were without regular skipper Kane Williamson who missed the match with a medical appointment leaving veteran Tim Southee to lead the side.
Opting to bat, New Zealand rode half-centuries from Devon Conway and Glenn Phillips to post 160 before being all out with two balls to go.
Arshdeep Singh removed Finn Allen in the second over to deal an early blow and then took a catch to send back Mark Chapman.
Phillips smashed 54 and Conway made 59 but New Zealand lost the plot in the last five overs, losing their last eight wickets for a mere 30 runs.
Mohammed Siraj was the pick of the Indian bowlers claiming 4-17, while fellow seamer Singh returned 4-37.
Southee, who had claimed a hat-trick in the previous match, dismissed Rishabh Pant and Shreyas Iyer with successive deliveries and India slumped 60-4 in the seventh over.
Pandya, however, looked in great form and smashed 30 not out off 18 balls before rain intervened.
India rested regular captain Rohit Sharma and batters Virat Kohli and KL Rahul for the series.
After the opening match in Wellington was washed out without a ball being bowled, India prevailed by 65 runs in the second match at Mount Maunganui, which proved decisive in the end.
The teams will now clash in a three-match one-day series beginning in Auckland on Friday.
