The tourists were 171-9 on day two after Jadeja bowled Matt Henry for his fourth wicket of the innings at the stroke of stumps on a sharply turning track in Mumbai.
Ajaz Patel was batting on seven after he took five wickets to help bowl out India for 263, with the hosts taking a slender lead of 28.
New Zealand won the first two matches to seal their first-ever Test series triumph in India and now eye a 3-0 sweep.
"To be honest it's (the pitch) turning quite sharply," Ajaz told reporters. "I suppose inconsistent in terms of how much it does turn and bounce. It has started to turn from both ends and from a batting perspective that can be a challenge."
India's Shubman Gill, who hit 90, said: "It's all about one good partnership when you are chasing a total which is around 150-160. If you have one partnership about 70-80 runs the match is done and dusted."
Will Young stood out with his 51 before being caught and bowled by Ashwin, who took three wickets with his off-spin.
Rohit Sharma arranges the field placement as Will Young (L) watches. AFP
Fast bowler Akash Deep struck in the first over when he bowled skipper Tom Latham for one to keep up India's fight after Gill's knock.
Washington Sundar sent back Devon Conway for 22, and Ashwin dispatched Rachin Ravindra for four, as New Zealand slipped to 44-3. But Young and Daryl Mitchell put on 50 runs to resist the Indian attack, which hit back through Jadeja's left-arm spin.
Jadeja got Mitchell out for 21, with Ashwin taking a stunning catch while running backwards from mid-off, and soon the bowler dismissed Tom Blundell.
Gill having fun
Earlier Ajaz's heroics — at the same venue where he claimed all 10 wickets in a Test innings in 2021 — helped end India's reply to their 235 in the second session. Gill and overnight partner Rishabh Pant, who hit 60, stood out as they counter-attacked in a stand of 96 while Sundar smashed an unbeaten 38 off 36 balls.
"I was honestly trying to have fun in the middle even if it was difficult — enjoy the difficult moments," said Gill. "If I was to put too much pressure on myself then I am losing on the art of batting."
Off-spinner Glenn Phillips struck soon after lunch to send back Jadeja, caught at slip for 14. Ajaz got Sarfaraz Khan caught behind for a duck in the next over.
Gill took on the spinners with boundaries in the next few overs before Ajaz had him caught at slip.
Sundar and Ashwin took India into the lead before Ajaz got Ashwin out for his sixth career five-wicket haul.
Resuming on 86-4, the batsmen came out aggressively as India look to avoid an embarrassing whitewash on home soil.
Gill survived a reprieve on 45 by substitute fielder Mark Chapman, who failed to hang on to a high catch off Phillips' bowling while running in from long-on. Gill reached his fifty with a single and Pant kept up the attack with regular boundaries to raise his half-ton in 36 balls.