Needing 107 runs on the final day, New Zealand reached their target just before lunch for the loss of two wickets after negotiating a tense final session.
New Zealand have only beaten India twice before on Indian soil, with the most recent occasion being 1988.
First innings centurion Rachin Ravindra was unbeaten on 39 and Will Young reached 48, with the pair putting on an unbeaten 75 for the third wicket following the early loss of openers Tom Latham and Devon Conway.
Victory was set up on the first day's play, when India were skittled for 46 - the lowest innings total by any team against New Zealand.
India revived their hopes of a fighting win when they scored 462 runs in their second innings, giving them a 106-run lead.
The crowd at M Chinnaswamy Stadium came alive when captain Latham departed on the second ball of the day, trapped LBW by Jasprit Bumrah for a duck.
Devon Conway fought gamely in difficult conditions, with India's seamers gaining considerable movement, but he eventually fell for 17 off 39 balls - also leg before wicket to the dangerous Bumrah.
Ravindra - who crafted a brilliant 134 in the first innings - immediately went on the front foot upon arrival at the crease, striking Bumrah for boundaries on two of his first three deliveries.
Young gained confidence and the pair advanced methodically towards the total in a 15.1 overs stand - snaring victory soon before the scheduled lunch break.
Time was also critical for the pair, with rain forecast for later in the day, with the potential to deny the tourists their slice of history.
Latham told Sky Sport his team had to fight all the way.
"Obviously the first two innings of the game we set the game up beautifully but we certainly knew that India were going to come back in that third innings and they put us under pressure for a long period of time.
"Then the way the seamers came back with that new ball, they came back and asked the right questions and got the result. So a fantastic performance.... it was nice to only be chasing a hundred."
Indian skipper Rohit Sharma said his team would carry out some soul-searching following their miserable first-innings score.
"I wasn't expecting that we'd be bowled out for under 50. But you have to give credit where it's due - New Zealand bowled pretty well and challenged every corner of our bat," he said.
"We failed to respond to that and that set us back in the game a little. Games like this can happen but we have to understand what we did well in this game as well."
The second match of the three-test series starts in Pune on Thursday, with New Zealand now eyeing the possibility of a first-ever test series win on Indian soil.
They can expect to face stung opponents, with India having won their previous 18 home test series and sure to come under pressure from home supporters and media.
