Jadeja (5/73) dismissed three batsmen in one over for a five-wicket haul while R Ashwin (4/93) grabbed four wickets as the Indians spun out New Zealand early into the second session.
At draw of stumps, India were 159/1 in their second innings which gave them an overall lead of 215. India lost KL Rahul (38) before Murali Vijay (64) and Cheteshwar Pujara (50) scored briskly to put the hosts in command.
While the Kiwis struggled to negotiate spin on a deteriorating track, the Indians were hardly troubled in their second innings with Pujara and Vijay going for their shots.
The Kiwi bowlers toiled without success post tea as Vijay and Pujara raised an unbeaten 107-run stand for the second wicket, scoring all these runs in the last session.
Vijay's knock came off 152 balls with seven shots to the ropes and one over it while Pujara struck eight boundaries in his 80-ball stay so far.
Leg-break bowler Ish Sodhi had Rahul caught at first slip in the fag end of extended second session. The umpires called for tea break, immediately after Rahul's wicket.
After starting the day in position of strength the New Zealanders could add only 100 runs to their overnight total with Ashwin and Jadeja exposing visitors' shortcomings in handling spin.
The Indians too had faltered in their first innings from being 154 for one to 318 all out.
Jadeja ran through the lower order in the post-lunch session by dismissing Mark Craig (2), Ish Sodhi (0) and Trent Boult (0) and Ashwin closed the innings by catching BJ Watling (21) in his follow through as the Kiwis lost last five wickets for just 24 runs in 10.5 overs.
Luke Ronchi (38), who had warmed up for the series with a century in the Delhi practice game, went about his business fearlessly as he took the fight to the rival camp.
He drove and cut Ashwin to boundary ropes with confidence and steadied the Kiwi innings along with Mitchell Santner (32), who was sent ahead of the more-experienced BJ Watling.
They added 49 runs for the fifth wicket before Jadeja trapped Ronchi with a full length delivery.
Ronchi, who hit six boundaries in his 83-ball innings, tried to sweep but the delivery was too full for that shot.
That was Ashwin's first ball of the session which turned from middle and leg, took and edge, lapped up by Wriddhiman Saha.
Jadeja was on verge of a hat-trick with wickets of Kraig and Sodhi in successive balls but Boult denied him the milestone before handing a catch to forward short-leg fielder in the last ball of the over.
Ashwin provided India the first breakthrough by trapping Tom Latham (58) in the fifth over of the day. It was a straight ball that struck the pads of Latham, who lunged forward to block it. The Kiwis had added only seven runs to their overnight total at that time.
With that dismissal, the 124-run partnership between Williamson and Latham ended.
The Kiwis had hardly settled when Jadeja, who was getting good turn and bounce, sent back Taylor, who also tried to defend on the front foot but was caught in line.
Ashwin and Jadeja had bowled testing 10 overs in tandem when India captain Virat Kohli brought part-timer Murali Vijay into the attack to bowl his off-breaks, replacing Jadeja.
Perhaps the motive behind the move was that New Zealanders' footwork was cagey against off-spin and a left-hander in Santner was also at the crease.
However, the pressure, which was created by Ashwin and Jadeja got released.
In the morning Ashwin, who had 3 for 63 in the first
While Ashwin ended up with three victims, Umesh Yadav was the best bowler from the home side with superb figures of 4 for 32, his best at home and second-best overall after the 5 for 93 he took against the same opponents in Perth in January 2012 on his debut.
However, the Indians' delight at ending the Australian innings, which had bloomed due to Starc's counter-hitting last evening, was nipped in the bud when they lost three of their top scoring batsmen within the first 15 overs.
Hazlewood removed in-form opener Vijay for 10 with a ball that moved in a shade and took the outside edge as he poked at it in the seventh over of the innings.
He first packed off another in-form batsman, Cheteshwar Pujara (6), with an unplayable snorter that the batsman gloved to wicketkeeper Matthew Wade.
The lanky pacer then struck the biggest blow of the morning by removing Kohli for a second-ball duck through a catch to the slip cordon as he drove away from body.
The Indian captain was dismissed for a duck for the first time at home and for only the fifth time in his 54-Test career. He had averaged 80 plus by tallying 1457 runs in the last 13 Tests.
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