Spinner Ajaz Patel claimed five wickets as New Zealand bowled out India for 263, with the hosts taking a lead of 28 on day two of a keenly fought third Test on Saturday.
New Zealand were 26-1 at tea after Akash Deep bowled skipper Tom Latham for one in the first over of the visitors' second innings at Mumbai's Wankhede Stadium.
Devon Conway, on 15, and Will Young, on eight, were at the crease at the break.
Shubman Gill top-scored for India with 90 before New Zealand ended the home side's reply to their 235 in the second session.
Gill and overnight partner Rishabh Pant, who hit 60, counter-attacked in a stand of 96 and Washington Sundar hit an unbeaten 38 off 36 balls.
Glenn Phillips struck soon after lunch to send back Ravindra Jadeja, caught at slip for 14.
Mumbai-born New Zealander Ajaz got Sarfaraz Khan caught behind for a duck in the next over.
Gill took on the spinners with boundaries in the next few overs before left-arm spinner Ajaz had him caught at slip.
Sundar, a left-handed batsman, and Ravichandran Ashwin took India into the lead before Ajaz got Ashwin out for his sixth career five-wicket haul.
Ajaz claimed all 10 wickets in an innings in a 2021 Test at the same venue.
Resuming on 86-4, the batsmen came out aggressively as India look to avoid an embarrassing 3-0 series whitewash on home soil.
Gill survived a reprieve on 45 by substitute fielder Mark Chapman, who failed to hang on to a high catch off Phillips' bowling while running in from long-on.
Gill reached his fifty with a single and Pant kept up the attack with regular boundaries to raise his half-ton in 36 balls.
New Zealand won the first two matches to seal their first-ever Test series triumph in India.