Replying to India's mammoth 557 for five declared, New Zealand were in a steady position going into lunch at 125 for one but a sudden collapse in post-lunch session saw them lose four wickets between 43rd and 49th over leaving them reeling at 148 for five.
Having provided the only breakthrough of the morning session, Ashwin (4/65) got three of these wickets and also also had a hand in Martin Guptill's (72) run-out.
Kiwis lost another wicket after Jimmy Neesham (37 batting) and BJ Watling (23) added 53 runs for the sixth wicket. Watling was sent packing by Ravindra Jadeja.
New Zealand need another 142 runs to avoid follow-on.
Ashwin first sent New Zealand captain Kane Williamson (8) and also accounted for the out-of-form Ross Taylor and Luke Ronchi, who both went without scoring. He had also sent opener Tom Latham (53) back 15 minutes before lunch.
He also figured in the run-out of the confident-looking Guptill who was unfortunately caught out of the crease as Ashwin got the tip of his finger to deflect a shot from Ronchi on to the non-striker's end stumps.
The quick fall of these four wickets left New Zealand in a deep hole before they recovered through the partnership between Neesham and Watling to reach 200.
On a song with the ball since dismissing Latham, the off spinner induced the visiting team s captain to cut a spinning-in ball and chop it onto the stumps.
Taylor and Ronchi were both deceived by the ball which did not turn much to be caught at slip off the outside edge while defending by Ajinkya Rahane, the latter off the wicket-keeper's thigh pad, so that Ashwin picked up three wickets in 13 balls.
In between the latter two dismissals was the run out of Guptill, who struck 2 sixes and ten fours to look well in control, as New Zealand suffered a steep slump from 134 for 1 to 148 for five within the space of five overs.
At the fall of Ronchi, Ashwin sported superb figures of 4 for 44 in 18.1 overs and at tea it read four for 65.
New Zealand had lost four wickets in 13 overs for 34 runs in the first hour's play after lunch when Ashwin was on a purple patch.
Jadeja was not given a spell in this successful period for the home side as captain Virat Kohli trusted his medium pacers to bowl from one end while utilising Ashwin from the other.
The left arm spinner was brought on to replace the off spinner just past the third hour of play after he had bowled with great success non-stop on either side of the lunch interval.
Jadeja succeeded in removing Watling who too, like some of his teammates, offered a defensive bat to a ball that spun very little and edged to Rahane who completed his third catch of the innings to add to his brilliant knock of 188.