Chasing a daunting 434, the Kiwis were 93 for four at stumps on day four and on Monday lost the remaining six wickets for just 143 runs.
India notched up a mammoth 197-run win in their historic 500th cricket Test to assert their supremacy in home conditions, bowling out New Zealand for 236 to take a 1-0 lead in the three-match series here today.
After setting the Kiwis a rather improbable target of 434, the Indian bowlers, especially senior off-spinner Ravichandran Ashwin (6/132), revelled in home conditions to rip through the visiting team's line-up with more than a session to spare.
Beginning the day at a precarious 93 for four, overnight New Zealand batsmen Luke Ronchi (80) and Mitchell Santner (71) frustrated the Indian bowlers for only about 20 overs by adding 65 runs in the morning session but that only delayed the inevitable.
The Black Caps raised hopes of a fight till the time Ronchi and Santner held fort but it all went downhill for them when the former played an extravagant slog sweep off Ravindra Jadeja, handing India the opening.
Ashwin (6/132) completed his fifth six-wicket haul by adding scalps of Santner and Ish Sodhi (17) and Neil Wagner (0) to his three wickets grabbed yesterday when he had become the second-fastest man to take 200-Test wickets.
Jadeja took one wicket today while Mohammed Shami, who got some reverse swing, scalped two batsmen in the morning session.
Ashwin grabbed the remaining three wickets in the post-lunch session to complete India's victory, their 88th on home turf.
Both the teams will now travel to Kolkata, where the second Test is scheduled to begin from Friday.
The Kiwis struggled to negotiate a slow track, which offered occasional turn and bounce but improved as the match wore on.
The visitors, chasing a mammoth 434-run target, had a daunting task at hand to bat out three full sessions on the fifth day, which has always been a big challenge for tourists in India.
With this win, India have taken their unbeaten tally on home soil to 11 matches. New Zealand, on the other hand, have remained without a win on the Indian soil since 1998.