Shubman Gill had earlier on Saturday just missed out on a century while Rishabh Pant scored a quick-fire 60 as India made 263 in response to New Zealand's 235.
Then the home spinners reduced the tourists to 9-171 by stumps, giving India the platform to ensure they don't get swept 3-0 in the series.
Glenn Phillips made a brisk 26 for the visitors before being dismissed. (AP PHOTO)
Ajaz Patel was unbeaten on seven after Will Young's battling 51 in testing conditions, as New Zealand wiped out a slender 28-run deficit and built up a lead of 143 at the end of another sweltering day at the Wankhede Stadium.
Ashwin, who took 3-63 and shared seven wickets with spin partner Jadeja (4-52), hoped the hosts would not be staring at a big target when they eventually bat on Sunday.
"We should wrap it up with one or two runs here and there. Every run saved in this innings is going to be critical for us when we chase it down," Ashwin said.
"It's not going to be easy and we have to bat really well.
"I expected a lot more bounce and speed from this Mumbai pitch. It has been quite slow, which is surprise for me ... It's not a typical Mumbai pitch."
Akash Deep put New Zealand under some early pressure in their second stanza when he castled skipper Tom Latham for one and Washington Sundar snared opener Devon Conway for 22 at the start of an absorbing final session.
Ashwin had Rachin Ravindra stumped for four to leave New Zealand in trouble, before pulling off a fine catch that got rid of Daryl Mitchell for 21.
Jadeja dismissed Tom Blundell for four, while Ashwin snared Glenn Phillips for 26.
Young looked to take the game away from India with a patient effort but spooned one back to Ashwin and Matt Henry was castled by Jadeja for 10 on the last ball of the day.
Earlier, Gill, having ridden his luck, edged spinner Patel to slip and departed for a fluent 90.
India's Shubman Gill sweeps during his fluent 90, his seventh Test half-century. (AP PHOTO)
Patel finished with 5-103, with Washington left unbeaten on 38.
A late collapse on Friday meant India resumed on 4-86, and Pant eased early nerves by hitting Patel for three boundaries in the opening over before scooping one for four a little later.
Gill was given a reprieve on 45 while looking to charge Phillips midway through the first session, when substitute Mark Chapman dropped a simple catch running in from the deep, and he went on to seventh Test half-century.
Pant, who hit Patel out of the park a couple of times and pulled Phillips for a four, reached his 50 in only 36 balls, the fastest by an Indian against New Zealand.
Sodhi trapped Pant lbw to end the fifth-wicket partnership for 96, Jadeja fell early in the second session for 14 when Phillips forced him to edge one to slip, and Sarfaraz Khan was caught behind off Patel for a four-ball duck.
An exciting finish could be in prospect with India far from out of the woods as they seek to avoid a similar calamity to 2000, when they lost 2-0 to South Africa, the last time they were blanked in a home series.
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