A defiant ninth-wicket partnership between Wriddhiman Saha and Bhuvneshwar Kumar helped India extend their lead and set a massive target for New Zealand.
New Zealand batsmen set out in their chase of 376 to deny India a 2-0 lead in the series.
Tom Latham and Martin Guptill survived a tricky 16-over period before Lunch to keep all wickets intact at the break.
Wriddhiman Saha scored his second fifty of the Test to power India ahead.
Martin Guptill fell right after Lunch but Tom Latham led New Zealand's fight.
Ravindra Jadeja broke the defiant stand between Tom Latham and Henry Nicholls, dismissing the latter for 24.
Ashwin unfurled his deceptive drifter and trapped Ross Taylor in front of the stumps cheaply.
Tom Latham's vigil was brought to an end by Ashwin, as the New Zealand was out caught behind.
Mohammed Shami got the ball to reverse and dismissed Mitchell Santner and BJ Watling in the space of two overs to reduce New Zealand to 156 for 6.
Ravindra Jadeja celebrates the wicket of Luke Ronchi
Wriddhiman Saha (left) was awarded the Man of the Match for his twin unbeaten half centuries and excellent wicketkeeping.
New Zealand were bowled out for 197 in the second innings, handing India a 178-run win. With the win, India have taken an unassailable 2-0 lead in the three-match series and reclaimed the No 1 spot in the ICC Test Rankings.