Ashwin's 6-54, his second six-wicket haul in the match, and left-arm spinner Pragyan Ojha's 3-48 bundled the visitors out for 164 in their second innings, giving India a 1-0 lead in the two-test series.
The 25-year-old Ashwin, who took six for 31 in New Zealand's first innings, grabbed 12 wickets for 85 runs in the match, the best by an Indian bowler against New Zealand in a test.
The Indian spin duo once again shared nine wickets as they had done in the first innings when New Zealand were bowled out for 159.
"This performance is very satisfying," India captain Mahendra Singh Dhoni said during the presentation ceremony.
"There was a bit of turn but it was a wicket where you had to keep coming at the batsmen, bowling in the right areas.
"With the batsmen deciding 'I'm not going to take you on', it was very difficult to get them out. Bowlers were patient, that was very important.
"We were also battling the weather because we never knew how many overs we'll get. That was the main reason why we wanted the opposition to follow on."
After a barren, rain-shortened morning session, India picked up five wickets in the afternoon session and completed the formalities soon after returning from the tea break.
Resuming on 41 for one, Brendon McCullum (42) and Kane Williamson (52) batted resolutely in the first session to deny the hosts a breakthrough.
New Zealand captain Ross Taylor, who was clean bowled trying to shoulder arms to Ashwin, said the team needed to find a way to handle the off-spinner better.
"We showed in periods that we could survive. The way Kane and Brendon batted showed rest of the batters that it could be done," Taylor said.
"We could not put enough pressure on the Indian bowling line-up. We need to learn how to play Ashwin and... play him a lot better.
"We will be a lot better side come Bangalore."
Umesh Yadav, the only Indian paceman to grab a wicket in the first innings, got McCullum lbw to end the second-wicket stand of 72.
McCullum was unhappy with the decision as he thought the ball had touched his bat before hitting the pad and trudged off shaking his head and muttering to himself.
The dismissal opened the floodgates as New Zealand lost their last seven wickets for just 26 runs.
Ojha got his second wicket of the innings by bringing Williamson's resistance to an end after the batsman had completed his half-century.
The start of play was delayed for the third day in a row due to morning showers. Most of Saturday's play was washed out due to a heavy downpour.
The second and final test of the series will be played in Bangalore from Friday.