Resuming day four on 159 for one, India's positive batting over two sessions yielded 218 runs before home captain Virat Kohli declared on 377 for five at Kanpur's Green Park Stadium.
Needing to pull off the biggest successful chase in the history of Test cricket for an unlikely win, New Zealand wobbled early and were reeling at 93 for four at stumps, still trailing the massive target by 341.
Luke Ronchi (38) and Mitchell Santner (eight) will return on Monday hoping to resist or at least delay India's victory march in the first match of the three-Test series.
Ravichandran Ashwin (3-68), who dismissed four frontline Kiwi batsmen in the first innings, proved once again why he is considered India's premier spinner.
The off-spinner had Martin Guptill caught at silly point for a duck with the first ball of his second over and trapped Tom Latham (two) leg before with the fifth.
Ashwin claimed his 200th Test wicket when he dismissed New Zealand skipper Kane Williamson (25) leg before.
Ross Taylor (17) then ran himself out to compound the tourists' misery.
Earlier in the morning, Williamson attacked with spin from both ends and his slow bowlers did not let him down.
Left-arm spinner Santner trapped Murali Vijay lbw for 76, ending the right-hander's second century-plus stand of the match with Cheteshwar Pujara (78).
Kohli did not look comfortable against Santner but it was off-spinner Mark Craig who dismissed the Indian skipper when he attempted a slog-sweep that found Ish Sodhi at deep midwicket.
Sodhi struck the next blow with a beauty of a delivery that spun from outside leg, took the outside edge of Pujara's bat and nestled into Ross Taylor's waiting hands at slip.
Ajinkya Rahane (40) missed his fifty but Rohit Sharma and Ravindra Jadeja smashed unbeaten half-centuries and featured in an unbroken century stand to virtually bat New Zealand out of the contest.
Sharma, who made 35 in the first innings, remained unbeaten on 68.
Jadeja, who scored 42 not out in the first innings and claimed five wickets with his left-arm spin, continued to have a good match.
The southpaw clobbered three sixes en route to his second Test fifty before breaking into a bat-twirling celebration.