India set New Zealand 284 runs after a lower order fightback from the Kiwi demolition of the hosts' top batters on the fourth day of the first Test in Kanpur on Sunday.
Ravichandran Ashwin then took the first Black Caps wicket in the day's penultimate over when Will Young fell leg-before for 2 in unfortunate circumstances.
Young, who scored 89 in the first innings at Kanpur, failed to review an umpire's call before the clock ran out -- and the Ashwin delivery was missing his stumps.
The final day will resume with Tom Latham on 2, and spinner William Somerville yet to score.
Shreyas Iyer -- the first Indian to score a century and a fifty on their debut -- bolstered his side's second innings with 65 after key partnerships with Ashwin and Wriddhiman Saha, who was unbeaten on 61 runs when the hosts declared at 234-7 after tea.
India had started in a commanding position with a 63-run lead at 14-1, after Axar Patel's five-for wrapped up the Black Caps innings at 296 on Saturday.
But New Zealand's Kyle Jamieson and Tim Southee turned the tables with crucial wickets to leave India reeling at 51-5.
Iyer, who scored his maiden Test century in the first innings, helped pull India out of trouble after a 52-run partnership with Ashwin, who left the pitch after chopping a rising Jamieson delivery.
Jamieson had triggered the Indian top order collapse with Cheteshwar Pujara's wicket for 22 in the 12th over of the innings.
India then lost stand-in captain Ajinkya Rahane at 41 after he fell leg-before to left-arm spinner Ajaz Patel.
Southee returned for an inspired bowling spell and took two wickets in his maiden over.
The veteran pacer first removed opener Mayank Agarwal for 17 after he edged to Tom Latham at second slip.
One ball later, Southee took all-rounder Ravindra Jadeja leg-before for a duck to leave India 51-5.
A stubborn lower order performance on an uneven surface saved the hosts further embarrassment as they reached 167-7 at tea.
"The idea was to play session by session," Iyer said.
Coach Rahul Dravid "said that I need to stay out in the middle for as much as possible and build on the score," he added.
Iyer built a 64-run partnership with Saha before falling to Southee, who took three wickets in his 22 overs for 75 runs.
Saha then played a gritty innings with Axar Patel, unbeaten on 28, to stitch an unfinished 67-run partnership before the Indian declaration.
Iyer said the hosts had the "spin power" to keep the visitors under pressure for the final day.
Jamieson, who took three wickets in the second innings, also became the fastest Kiwi to 50 wickets in his ninth Test, beating Shane Bond's record by three matches.
Patel took another wicket but the other two New Zealand spinners -- Rachin Ravindra and William Somerville -- went wicketless.