New Zealand were comfortable on 125-1 at lunch after Martin Guptill and Tom Latham featured in the first century opening stand of the series but Ashwin (4-61) came alive in the second session to leave the tourists staring at a possible follow-on.
Jimmy Neesham (37) and Mitchell Santner (six) were in the middle at the tea break with New Zealand still trailing India, who hold an unassailable 2-0 lead in the series, by 341 runs.
Resuming day three on 28-0, Latham (53) and Guptill (72) were positive in their mindset and decisive in their footwork against the Indian spinners, even though both had their share of luck.
Guptill was on 21 when he edged paceman Mohammed Shami but Ajinkya Rahane spilled a catch at gully that came at stomach height.
Guptill landed in India with question marks over his place in the test squad and scores of 21, 0, 13 and 24 in the last two tests did little to silence the doubters.
However, the right-hander confirmed he had rediscovered his touch when he blasted Ravindra Jadeja for his second six and hit the left-arm spinner for a boundary to bring up his 17th test fifty.
Latham, too, brought up his fifty with a boundary off the spinner but was lucky to be given the benefit of the doubt by the umpire when he dragged the ball onto his boot while sweeping and was caught by Rahane at slip.
Ashwin, however, ensured Latham's stay did not last much longer when he sent down a teasing off-break in the next over which the batsman played across and spooned a leading edge back to the bowler.
Wickets started tumbling after lunch with skipper Kane Williamson (eight) pushing at an Ashwin delivery with an angled bat, only to drag the ball on to his stumps.
Ross Taylor did not bother the scorers and was caught by Rahane at slip, the bowler-fielder duo combining again to send back Luke Ronchi (0) and BJ Watling (23).
Ashwin had a hand in Guptill's dismissal too, when the ball brushed his finger as he stooped to stop a Ronchi drive before it hit the stumps with the opener backing up way too far at the non-striker's end.
(Reporting by Amlan Chakraborty in New Delhi; Editing by John O'Brien)
