Rohit Sharma slammed 65 and gave India a strong start.
Hamish Bennett went for 54 but bagged three wickets to pull things back.
Manish Pandey and Ravindra Jadeja gave India the late push to post 179 for 6.
Kane Williamson kept things alive in the chase with a superb 95 off 48.
With one needed off one, Ross Taylor was bowled by Mohammed Shami on the final ball.
Jasprit Bumrah had an off day as he first gave away 45 from four overs and then 17 runs in the Super Over after New Zealand failed to score 3 off 5 in the last over.
Needing 10 off last two, Rohit Sharma hit two sixes to seal the game for India. This is India's first T20I series win in New Zealand.