Needing 19 off the final over for the win, and with just one wicket in hand, India came agonisingly close to beating New Zealand's 314 all out off 50 overs.
Ravindra Jadeja belted 15 off the last over, with three wides from Corey Anderson adding to the final tally and leaving the scores level.
The Black Caps looked to have done enough to wrap up the five-match series after picking up the key wickets of Virat Kohli for six and captain MS Dhoni (50).
But Ravi Ashwin (65) and Jadeja resurrected India's hopes in putting on 85 for the seventh wicket, with Jadeja's unbeaten 66 off 45 balls proving everything but a match-winner.
Anderson couldn't deliver with the bat, managing only eight, but proved lethal with the ball and finished with figures of 5-63, including the key scalps of openers Rohit Sharma and Shikha Dhawan.
Hamish Bennett, making a long-awaited return to international cricket from injury, provided superb back-up with 2-41 off his 10 overs.
Earlier, New Zealand got off to a flying start after losing the toss for the third successive game.
Martin Guptill provided the backbone of the innings with 111 - his fifth ODI century - and the opener's 153-run partnership with Kane Williamson (65) set a new record for the second wicket against India.
But a flurry of dismissals slowed the New Zealand run-rate after they initially looked on track to comfortably score 330.
The home side were 1-189 in the 33rd over but then lost four wickets in 36 balls for 41 runs, with Williamson bowled by Mohammed Shami for 65 and Guptill caught in the deep by Ajinkya Rahane off Jadeja.
Five more wickets fell in the last 7.2 overs, Luke Ronchi's 20-ball 38 the highlight in tandem with a breezy 27 off 23 balls from Tim Southee at the tail-end of the innings.
New Zealand won the first two ODIs, with the fourth scheduled for Seddon Park in Hamilton on Tuesday and the fifth in Wellington on Friday.