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The victory was New Zealand's 100th since they began playing Tests in 1930. It came against a side unbeaten in their last nine Tests and one that had won all seven of their previous World Test Championship matches.
The return of New Zealand pace spearheads Tim Southee and Trent Boult and the introduction of 2.03m-tall debutant Kyle Jamieson proved crucial against the tourists, also the world's top-ranked Test side.
They exploited a tricky Basin Reserve wicket to bowl India out for under 200 twice with Jamieson's height and delivery angle providing plenty of problems.
Virat Kohli's side lost their final six wickets for 47 runs as Boult and Southee combined to rip through the lower middle order and then finish off the tail. Southee completed his 10th five-wicket haul in Tests with 5-61.
Tom Latham finished seven not out while Tom Blundell was on two as they knocked off the nine runs needed for victory in just 10 balls after the visitors were dismissed for 191 in their second innings.
"It was a great win and to beat a quality Indian side is very pleasing," said Southee, named man of the match for figures of 9-110.
Jamieson's emergence as a pace bowling threat who can also bat - he scored 44 in New Zealand's first innings - could create some selection issues with Neil Wagner coming back for the second game starting at Hagley Oval on Saturday.
"The performance across the board was a very good one," captain Williamson told reporters. "We know we have good days and bad days but guys came back ... and really focused on the areas we needed to move this team forward.
"That was the most pleasing performance."
Kohli held his hand up and conceded his side had not been good enough but added that it was only one defeat.
"Some people might want it to be the end of the world, but it's not. For us it's a game of cricket," he said. "We lost and we move on, we keep our heads high."
On the plus side for the visitors, Ishant Sharma took 5-68 in New Zealand's first innings after returning from a torn ankle ligament and was especially effective when they reduced the hosts to 225-7 on day three.
India had started the day on 144-4, still 39 runs behind, and their hopes of setting a decent target rested with the last recognised batsmen - vice-captain Ajinkya Rahane and Hanuma Vihari - seeing out the first session unscathed.
But Boult, in his second over, enticed an edge from Rahane when he was on 29 and wicketkeeper BJ Watling completed an easy catch. Vihari had not added to his overnight score of 15 when he was bowled by Southee in the next over.
Southee then had Ravichandran Ashwin trapped in front for four to reduce India to 162-7.
After taking India past New Zealand's first innings with a four off Jamieson, Ishant (12) was then trapped in front by Colin de Grandhomme in the next over before Southee had Rishabh Pant caught by Boult at fine leg for 25.
Southee wrapped up the innings on the sixth delivery with the second new ball when Jasprit Bumrah was caught in the slips for a duck.
Kohli said being dismissed for 165 in the first innings had been crucial.
"As a batting unit we take pride in being competitive, but we weren't," he said.
"Even scoring 220-230 would've been good. That first innings put us behind and then the lead put us under more pressure."Scoreboard
India 1st innings 165 (A. Rahane 46; K. Jamieson 4-39, T. Southee 4-49)
New Zealand 1st innings 348 (K. Williamson 89, R. Taylor 44, K. Jamieson 44; I. Sharma 5-68, R. Ashwin 3-99)
India 2nd innings (overnight 144-4)
P. Shaw c Latham b Boult 14
M. Agarwal c Watling b Southee 58
C. Pujara b Boult 11
V. Kohli c Watling b Boult 19
A. Rahane c Watling b Boult 29
H. Vihari b Southee 15
R. Pant c Boult b Southee 25
R. Ashwin lbw Soutee 4
I. Sharma lbw de Grandhomme 12
M. Shami not out 2
J. Bumrah c sub (Mitchell) b Southee 0
Extras (w2) 2
Total (all out, 81 overs) 191
Fall of wickets: 1-27, 2-78, 3-96, 4-113, 5-148, 6-148, 7-162, 8-189, 9-191, 10-191
Bowling: Southee 21-6-61-5, Boult 22-8-39-4, de Grandhomme 16-5-28-1, Jamieson 19-7-45-0 (2w), Patel 3-0-18-0
New Zealand 2nd innings
T. Latham not out 7
T. Blundell not out 2
Extras 0
Total (0 wickets, 1.4 overs) 9
Bowling: Sharma 1-0-8-0, Bumrah 0.4-0-1-0
Result: New Zealand won by 10 wickets
Toss: New Zealand
Umpires: Richard Kettleborough (ENG), Aleem Dar (PAK)
TV Umpire: Michael Gough (ENG)
Match referee: Ranjan Madugalle (SRI)
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