The 25 year old was superbly backed by left-arm spinner Pragyan Ojha (3-44) as New Zealand were dismissed for 159 in their first innings at the stroke of lunch in reply to India's 438.
The hosts, who enforced the follow-on after taking a 279-run lead, reduced New Zealand to 41 for one in the second innings at stumps on a day when only 37.3 overs were bowled.
Opener Brendon McCullum was unbeaten on 16 and Kane Williamson was three not out when play was called off after tea due to rain. New Zealand trail by 238 runs.
India opened the attack in the second innings with Ojha, who trapped opening batsman Martin Guptill (16) leg-before in his eighth over to claim his fourth wicket of the match.
"There is some bounce in the wicket and in the second innings, it has started turning more. We are confident that everything will fall in place for us," said Ojha.
"When it is turning, it's a bonus for you. Keep it there and let the batsmen make some mistakes. We have two more days to go and are confident with the way the guys are bowling and how everyone is approaching the game."
The tourists earlier added 53 runs to their overnight total of 106 for five after the match started an hour late due to a wet outfield, with Ashwin taking three wickets and Ojha and pacer Umesh Yadav one each.
It was Ashwin's third haul of five or more wickets in an innings in seven Tests.
James Franklin, unbeaten on 31 on Friday, was the lone batsman to defy the Indian attack, top-scoring with an unbeaten 43 off 122 balls.
"Obviously there was always going to be some balls that were going to spin past me. I was just trying not to worry about them too much," said Franklin.
"Hopefully, we will rectify a few mistakes we made in the first innings and look to stay a lot longer at the crease. It's a pretty simple formula: just occupy the crease. They are obviously good spinners and done well in India. They have a lot of fielders around the bat. We can't let that worry us too much and we have to just concentrate on our own game."
India struck in the day's second over when Yadav trapped Kruger van Wyk leg-before for no score and then Ojha had Doug Bracewell (17) stumped to reduce the tourists to 141 for seven.
Ashwin got a wicket with his third delivery when he held a low return catch to his left to dismiss Jeetan Patel (10) before removing Trent Boult and Chris Martin off successive balls.
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