The hosts were five wickets away from a 1-0 series victory with New Zealand reaching 140 for five at close of play on day three, chasing an improbable target of 540 and with two days still remaining.
It was a much better showing with the bat from the world test champions who managed only 62 in their first innings in reply to India's 325 on a pitch offering plenty of assistance to the spin bowlers.
Ashwin, third on the list of most successful Indian test bowlers with 426 scalps, dismissed stand-in captain Tom Latham lbw before the players left the field early for the tea interval due to a technical problem with the Spidercam.
The wily 35-year-old returned to send back Will Young and Ross Taylor in the and reduce New Zealand to 55 for three.
Daryl Mitchell and Henry Nicholls shared a stand of 73 before left-arm spinner Axar Patel dismissed the former for an attacking 60.
The touring side's problems were compounded when Tom Blundell had a brain fade moment and was run out without scoring.
Nicholls survived nervous moments but remained unbeaten on 36 with Rachin Ravindra on two not out.
"Obviously, it's a very, very long road ahead but hopefully we can build some partnerships and play ourselves in," Ravindra said. "Two of them (Mitchell and Nicholls) showed us how to get it done and we will hope to build on it."
Earlier, the hosts declared on 276 for seven, ensuring their bowlers had enough time to dismiss New Zealand after the opening test in Kanpur ended in a thrilling draw.
Ajaz takes 14
A day after becoming only the third bowler to pick up all 10 wickets in a test innings, the Mumbai-born Ajaz Patel picked up four more to take his match tally to 14.
Left-arm spinner Ajaz finally received support from his slow bowling colleagues with part-time left-armer Rachin Ravindra picking up 3-56.
The New Zealand fast bowlers again failed to pick up a wicket but stopped the Indian batters from scoring freely with a tight line from around the wicket.
Resuming on 69-0, India openers Mayank Agarwal and Cheteshwar Pujara extended their partnership to 107 before Ajaz sent the former back.
Agarwal was out caught for a confident 62 to add to his first-innings score of 150. Pujara also looked set for a half-century but was out caught in the slips for 47.
Regular opener Shubman Gill, who did not bat in his usual position after suffering a blow to his elbow while fielding, also missed out on a half-century by getting out on 47 as did India captain Virat Kohli who made 36.
All-rounder Axar provided fireworks during his unbeaten 41, entertaining the Sunday crowd at the ground overlooking the Arabian Sea with four sixes and three fours in his 26-ball innings.
"As time is progressing, the track is slowing down a bit, so today, even when there was turn, the batter had enough time to go on the back-foot and negotiate it," Axar said.
"Only if the batter is making mistakes, then it's a bit difficult for him. We have been in umpteen situations like these and we know what needs to be done in these circumstances."
