Will Young and Rachin Ravindra put on an unbroken 75 to guide New Zealand to a first Test win in India for 36 years, by eight wickets, on day five of the rain-hit match in Bengaluru.
Chasing a tricky 107 on a rain-delayed day, New Zealand lost skipper Tom Latham without scoring and Devon Conway for 17 before Young and Ravindra guided the visitors to 110-2 in the first session and a 1-0 lead in the three-match series.
First-innings centurion Ravindra, a wristy left-hander, looked comfortable as he hit six fours in his 46-ball stay.
Young hit the winning boundary, one of his seven fours, as his New Zealand teammates exchanged hugs in the pavilion. He also hit one six, while facing 76 balls.
It was New Zealand's third Test victory in India after their first in 1969 at Nagpur and a second at Mumbai in 1988.
Latham, who took over from Tim Southee after New Zealand were swept 2-0 in a Sri Lanka Test series recently, marked his first Test as full-time captain with a resounding victory.
Fast bowlers Matt Henry and Will O'Rourke set up the win with 15 wickets in the match as India, having chosen to bat after the first day was lost to the weather, were skittled for 46 in their first innings, their lowest home Test score.
On Sunday, India fast bowler Jasprit Bumrah gave the home side a glimmer of hope when he removed Latham lbw with the second ball of the day and New Zealand yet to score.
Bumrah and new ball partner Mohammed Siraj then subjected the New Zealand batsmen to a testing start, with deliveries consistently going past the bat on a lively wicket that had been under covers earlier because of rain.
Bumrah struck a second time when he trapped the left-handed Conway lbw after a successful review to leave New Zealand 35-2.
But that was India's last breakthrough as Young and Ravindra guided their team to a famous win.
New Zealand had replied to India's first innings 46 with 402 then dismissed India for 462 in their second innings.
Sarfaraz Khan scored 150 and Rishabh Pant 99 as their 177 stand led India's second-innings fightback, but from 408-3 Henry and O'Rourke sparked a collapse as the hosts lost their last seven wickets for 54.
Ravindra, a Wellington-born batting all-rounder with roots in Bengaluru and his father watching from the stands, hammered 134 in New Zealand's first innings.
New Zealand came into the match without leading batsman Kane Williamson, who is recovering from a groin injury.
The second Test begins on Thursday in Pune. The third is on November 1 in Mumbai.