Both the New Zealand openers, Tom Latham (31) and Martin Guptill (24) looked steady in the middle while playing 16 overs before the break.
Latham hit four boundaries and looked solid in his innings even as Mohammed Shami tested him by bowling round the wicket.
Guptill, who has been short of runs in the series, looked anxious and survived a close LBW appeal on 6 when he was trapped in front of the middle stump but umpire Rod Tucker felt differently.
Ravichandran Ashwin was introduced in the ninth over but the track offered little turn and the Kiwi opening duo survived the first session of the fourth day.
Trailing 0-1 in the three-match series, the Kiwis have two more sessions and one full day tomorrow to save the Test and series.
Earlier resuming the day at 227 for eight, India were bundled out for 263 in 76.5 overs setting New Zealand an improbable target of 376. India were all out after Wriddhiman Saha struck his second half-century of the match, remaining unbeaten on 58.
For the Kiwis, Trent Boult (3/38), Matt Henry (3/59) and Mitchell Santner (3/60) were the pick of the bowlers.
Number eight Saha had also remained not out in the first innings with a stellar 54 and he followed it up with yet another impressive knock, his fourth fifty.
Resuming the day on 39, Saha was watchful to start with, getting himself adjusted to the variable bounce off the pitch.
Bhuvneshwar Kumar at the other end looked to go for his strokes. Stand-in skipper Ross Taylor brought in Neil Wagner in the 10th over of the day and the left-arm pacer gave the breakthrough by removing Bhuvneshwar (23) to reach to his 99th scalp in his 25th Test.
At the other end, Saha remained calm and finally got to his fifty in 113 balls.
He swept Santner for two delightful boundaries to get past the mark but he was left stranded with last man Shami becoming Boult’s third victim.
Shami fell to a low catch by Latham, who was in the slip cordon.
PTI