Ashwin (5-69), who had a 12-wicket haul in the Hyderabad Test that India won by an innings and 115 runs, bowled quite beautifully to fox the Kiwi batsmen into errors as the visitors ended the day at 232 for nine, just 244 runs in front with two more days left in the Test.
Earlier, the Indians were bowled out for 353, 12 runs behind the Kiwi first innings total after Virat Kohli (103) notched his second century in 10 Tests, but it was seamer Tim Southee who took the honours in the morning by claiming seven for 64, his best figures to date.
If the second new ball pegged the Indian innings, then Umesh Yadav struck two telling blows first up to open the doors for India while Ashwin, who had 3 for 24 in his third spell of eight overs, and left-arm spinner Pragyan Ojha (2-48) made further inroads into the Kiwi line-up.
India's chances hinge on their ability to counter the Kiwi seamers on the morrow while the weather, which has been sunny thus far, could also influence the course of the match that is nicely poised.
India enjoyed a fruitful post-lunch session when they delivered big blows to put New Zealand under pressure as the visitors lost four top-order batsmen after early strikes by Yadav, who dismissed openers Martin Guptill (7) and Brendon McCullum (23) in quick succession.
Guptill missed a full-toss to be bowled leg-stump in the first over after lunch while McCullum, who had started off in a bright fashion with a series of boundary hits, nicked to 'keeper Dhoni in the next over.
A dour Kane Williamson (13) and skipper Ross Taylor (35), the first innings centurion, then batted with defiance. However, the introduction of Ashwin saw the exit of Williamson who edged to lone slip fielder Virender Sehwag.
Taylor and Daniel Flynn (31) batted steadily until Taylor missed a sweep off Ojha and was trapped in front after a 42-run partnership. Soon after tea break, Flynn was dismissed as he guided Ashwin into the hands of Sehwag and the Kiwis were floundering on 140 for five.
However, little Kruger van Wyk (31), who was struck on the head and left fore-arm off consecutive deliveries from Yadav, and the tall James Franklin did a brief rescue act by adding 55 runs for the sixth wicket before Ashwin struck.
The tall off-spinner beat van Wyk's tentative bat to catch him plumb in front and later, drew Franklin out and had him stumped. Ashwin went on to take his fifth wicket by knocking back Tim Southee's stumps with one that turned and went in through the gate.
Towards close, Ojha trapped Bracewell in front to leave the Kiwis on 232 for nine at stumps.
Earlier, it was a sensational first hour's play that saw Southee coming up with a brilliant spell during which he took four wickets to pin India, overnight 283 for five, on the ropes.
The Kiwis wasted no time to take the second new ball and after a poor over by Trent Boult who conceded nine runs, Southee took over at the other end and carved his way through the Indian line-up.
In the meantime, Kohli, overnight 93, got to his century with two boundaries, but became first of Southee's four victims of the morning when he padded up to an in-coming delivery to be adjudged leg-before, thus ending a 122-run partnership with Mahendra Singh Dhoni (62).
In the following over, Southee claimed another leg-before decision, this time Dhoni playing down the wrong line. Zaheer Khan blasted a six off Southee, but the Kiwi got rid of the tail-ender with a snorter to have him caught behind and followed up with the wicket of Ojha, also snicking to the wicket-keeper in the same over.
Thereafter, the plucky Ashwin (32 not out), who has a Test century to his name, and Yadav (4) put on 33 runs for the last wicket before the latter fell to Boult as India fell 12 runs in arrears on the first innings.