Indore, October 10: Riding on Ashwin's brilliant figures of 6/81, India bowled out New Zealand for 299 thereby taking a healthy first-innings lead of 258 runs.
Virat Kohli, however, did not enforce the follow-on and India were 18 for no loss at stumps increasing their overall lead to 276 runs.
Veteran opener Gautam Gambhir was retired hurt on six after sustaining a shoulder injury going for a second run. At stumps, Cheteshwar Pujara (1) and M Vijay (11) were at the crease for the home team.
The day belonged to Ashwin, who used the bowlers' rough brilliantly by operating from the pavilion end to get his 20th five-wicket haul in his 39th Test.
The off-spinner, who grabbed 10 wickets in the series opener at Kanpur and four in the second match at Kolkata, also ran out two batsmen off his own bowling.
His spin partner Ravindra Jadeja accounted for the other two wickets in the visitors' innings.
With two days left in the game, India, who won the first two Tests by huge margins, looked well-poised to score another victory for a clean sweep of the series.
After the run-glut from captain Virat Kohli (211) and Ajinkya Rahane (188) on the first two days, it was the spin magic from Ashwin that enthralled the big crowd after the Kiwis had put on the first century stand of the series from either side for the first wicket.
Ashwin came up with a sensational post-lunch spell to grab four top order wickets, including three in the space of 13 balls, after New Zealand openers Martin Guptill (72) and Tom Latham (53) had made a great start by putting on 118 runs.
Ashwin sent back Latham, to break the opening partnership before lunch, and then dismissed captain and No 3 Kane Williamson (8) before adding the scalps of Ross Taylor (0) and Luke Ronchi (0) to trigger a top-order collapse from which New Zealand could not recover.
He then trapped the stubborn James Neesham on the sweep shot in front of the stumps for 71 and then brought the innings to an end by having last man Trent Boult caught in the deep.
He was also responsible for running out Guptill and Jeetan Patel by deflecting the ball on to the non-striker's end stumps, as the batsmen were caught out of the crease.
Left-arm spinner Jadeja, who claimed 2 for 80 by breaking two half-century partnerships down the order, gave Ashwin good support as the Kiwis folded up forty minutes before the scheduled close.
Ashwin, who shot past Ishant Sharma in the Indian Test bowlers' list of wicket takers, bowled a sensational post lunch spell to leave New Zealand in tatters.
The off-spinner, who got one wicket in the morning session, grabbed three more wickets besides running out well-set opener Guptill post lunch to leave the visitors gasping at 148 for five in the first hour before New Zealand made a partial recovery.
On a song with the ball since catching Latham off his own bowling off a leading edge, 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 Rahane, the latter off the wicketkeeper's thigh pad, so that Ashwin picked up three wickets in 13 balls.
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.
Jadeja then got into the act by ending the sixth wicket partnership that realised 53 runs between Neesham, who batted well against the spinners, and wicket keeper Brad Watling (23) by getting rid of the latter.
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 4 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.
