R Ashwin, who has been in brilliant form in the last few years, was once again India's hero as he claimed another five-wicket haul.
Chasing an impossible 434-run target, Kiwis were bowled out for 236 in 87.3 overs on the final day of the opening Test.
Resuming on 93/4, New Zealand were off to a good start in the first session where Mitchell Santer and Luke Ronchi, two unbeaten batsmen on Day 4 continued to frustrate the Indian team with a brilliant fifth-wicket stand. (Ravichandran Ashwin becomes fastest Indian to take 200 Test wickets)
Both the batsmen played some delightful strokes and India had to wait for quite some time for the first breakthrough in the morning session.
It came in the 58th over of the innings when Ronchi (80) perished while trying to hit Ravindra Jadeja for a six. He played against the spin and holed one towards point which was easily caught by Ashwin. The wicket also ended the 102-run stand for the fifth wicket.
Mohammed Shami then provided two quick breakthroughs when he dismissed BJ Watling (18) and Mark Craig (1) in consecutive overs.
After lunch, Ashwin claimed the remaining three wickets, including the wicket of Santner(71), and also completed yet another ten-wicket haul. (WATCH: Ravichandran Ashwin's magic ball that bamboozled Kane Williamson on Day 3)
India now lead the three-match series 1-0.
Ravindra Jadeja, who picked six wickets and scored valuable 92 runs, was adjudged Man of the Match.
The second Test is scheduled to be played from September 30 at Eden Gardens in Kolkata.