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.
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.
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.
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.
(REOPENS DEL 13)
In the morning session the Kiwis got off to a splendid start in chase of India's imposing total when Guptill and Latham put on 118 runs for the first wicket after resuming from the overnight 28 for no loss before Ashwin struck to end the partnership.
Both Guptill and Latham completed half centuries and also figured in the first century first wicket stand of the series when Ashwin nipped it by dismissing the left-handed Latham caught and bowled for 53.
Past the hour the right-hander, having grown in confidence, lofted Jadeja over the long off region for a six, his second of the knock after his lofted effort off Ashwin last evening. He also got past his previous highest knock of 24 in this series.
Latham, at the other end, edged Ashwin for a four and then struck the off spinner - when he bowled short - for two more fours in the 29th over to bring up the team's 100.
Latham survived a close call off Jadeja when he swept the left arm bowler only for the ball to take the inside edge and strike his boot on the full before ballooning to short gully fielder Rahane, but umpire Bruce Oxenford denied the Indians the wicket by turning down the concerted appeal.
However, the opener - who hit 7 fours in his 102-ball knock - did not last long as Ashwin - in the next over - accepted an easy catch off his own bowling off the leading edge as Latham tried to turn to leg a tossed up ball from the off spinner.
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