New Zealand started the bright and sunny day at a strong 152 for one but Indian spinners claimed three wickets in the space of 29 balls to send the visitors on the back-foot.
Ashwin removed both the overnight batsmen -- Tom Latham (58) and captain Kane Williamson (75) -- while left-arm spinner Jadeja got rid of dangerman Ross Taylor (0) and Luke Ronchi (38) as the Green Park pitch offered turn and bounce this morning.
Ronchi, who had warmed up for the series with a century in the Delhi practice game, went about his business fearlessly as he drove and cut Ashwin to boundary ropes with confidence.
With his fearless approach, he steadied the Kiwi innings along with Mitchell Santner (28 batting), who was sent ahead of much-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.
Since the final session was washed out yesterday, the match began 15 minutes ahead of the scheduled start today.
India took the new ball immediately after completing 80 overs but did not get success as the Kiwis trail by 80 runs.
Ashwin provided India the first breakthrough by trapping Latham 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 frontfoot but was caught in line.
After two quick blows, the Kiwis needed Williamson to stay but Ashwin saw the back of rival captain with a peach of a delivery that turned sharply from outside off to hit the stumps.
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 rationale 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. Further, introduction of paceman Umesh Yadav into the attack brought relief for the visitors, since handling the pace was much easier than the spin.