After dismissing India for their worst home total of 46 and making 402 in reply, New Zealand bowled out Rohit Sharma's side for 462 in the second innings on Saturday to lay the platform for a 1-0 lead in the three-match series.
Will Young (48 not out) and Rachin Ravindra (39 not out) remained calm under pressure after the loss of two early wickets to get the job done, earning New Zealand only their third win on Indian soil in 38 attempts going back to 1955.
"It was probably a good toss to lose," said New Zealand skipper Tom Latham. "The guys came out in that first innings, put the ball in the right areas for long periods of time and managed to get the results.
"The first two innings of the game... I think we set the game up beautifully. We knew India were going to come back in the third innings and they put us under pressure, but the way the seamers came back with the new ball... fantastic performance."
New Zealand got off to a wobbly start when play began after a rain delay, as new permanent skipper Latham was trapped lbw by Jasprit Bumrah on the second ball of the day for a duck with the tourists yet to score.
Bumrah and fellow pace bowler Mohammed Siraj made life hell for the New Zealand batsmen early in the day, as India looked to do what no team have done in Test history and win a match after conceding a first-innings lead of more than 350 runs.
Conway endured a couple of painful body blows, balls that whizzed past the blade and unfriendly glares from the bowlers amid jeers from partisan fans at the M Chinnaswamy Stadium before falling lbw to Bumrah for 17.
The wicket appeared to turn placid after the arrival of first innings centurion Ravindra, and the Wellington-born left-hander with roots in Bengaluru superbly steered New Zealand out of rough waters before Young hit the winning runs.
Rohit rued India's batting effort in the first innings but took the positives from their second stanza, where Sarfaraz Khan made 150 and Rishabh Pant hit 99 to gave them a fighting chance.
"It was a good effort in the second innings with the bat. We didn't bat well in the first, so we knew what was lying ahead. We wanted to get as many as possible. A couple of guys really stood out and it was great to watch that," Rohit said.
The series, which also features matches in Pune and Mumbai, is part of the World Test Championship.
India arrived in Bengaluru seeking a big win over the 2021 champions to improve their chances of making a third successive final in June 2025, but will now need to battle to keep alive their run of 18 straight home series victories since 2012.