New Zealand, trailing 1-0 in the two Test series, have a lead of 244 runs with their number 10 batsman Jeetan Patel (10) and number 11 Trent Boult (nought) at the crease.
Batsman Virat Kohli, who completed his second Test century in the morning, said India had enough time to chase down any target with two more days left in the match.
"When you have a lot of time you don't have to think about the target, you just play normal cricket," Kohli said.
"I don't think the wicket is doing so much that we have to be worried about the target that's been set.
"If we think to bat three-four sessions, we should be in a good position to win the match."
Ashwin, who took 12 wickets in the first Test at Hyderabad which the hosts won by an innings and 115 runs, and left-arm spinner Pragyan Ojha (2-48) got into the act after paceman Umesh Yadav (2-62) sent back the openers.
Right-hander Martin Guptill (seven) fell in the first over after the lunch break when he played a full toss from Yadav on to his stumps.
The paceman returned in his next over to dismiss the dangerous Brendon McCullum (23), who got a feather edge to wicketkeeper Mahendra Singh Dhoni behind the stumps.
Kane Williamson (13) was caught by Virender Sehwag at slip off Ashwin while Ojha dismissed first innings centurion Ross Taylor (35), out leg before trying to play the sweep.
Left-hander James Franklin (41) and wicketkeeper Kruger van Wyk (31) added 55 for the sixth wicket to extend New Zealand's lead before Ashwin ended the resistance by trapping Van Wyk lbw.
The New Zealand batting tail failed to wag after Franklin was out stumped, trying to dance down the wicket to the 25-year-old off-spinner.
Bob Carter, the New Zealand assistant coach, hoped their lead would be sufficient to press for a series-levelling victory.
"Now we have a handy lead and that has given us an opportunity in this game," Carter told reporters. "We have three young bowlers who are raring to go.
"We are going to go out there and look to a fresh guard. We are going to look to make the ball swing and get into a good enough position to press for a win."
Earlier, Tim Southee mowed down India's batting line-up with the second new ball to pick up a career-best seven wickets as the hosts were dismissed for 353 in their first innings.
Paceman Southee, 23, who took three wickets on Saturday, destroyed India's hopes of a first-innings lead with figures of seven for 64, the best by a New Zealand bowler in an innings in India, to give his team a slim 12-run lead.
"I really thought it was a wonderful performance. In this Test after being selected, he really paid back the selectors and I think he really bowled very well to the overnight batsmen today," Carter added.
"To get Kohli, who was on a hundred, the way he did was an outstanding piece of bowling.
"Seven wickets in India is not just something that he would remember, that's what we all would remember as well."
Kohli, unbeaten on 93 overnight, struck two boundaries to complete his second test century before he became Southee's first victim of the morning.
The 23-year-old Kohli (103) hit 14 fours and a six and added 122 runs for the sixth wicket with captain Dhoni.
He misread an incoming delivery from the right-arm paceman, who was drafted in for the second Test replacing pace colleague Chris Martin, and was caught plumb in front of the stumps.
Dhoni (62), who hit his 26th half-century in tests, was also out lbw to become Southee's 50th Test victim.
Zaheer Khan (seven) and Ojha (nought) were next to go, both edging Southee deliveries to wicketkeeper van Wyk in the same over.
Ashwin (32 not out), put down by Guptill on 13 off Southee, added 33 crucial runs for the last wicket with Yadav (four) to cut New Zealand's advantage.
But left-arm seamer Boult finally broke through the stubborn resistance by bowling out Yadav to end the innings.