Boult, back in Test cricket after breaking a hand during the Boxing Day Test in a disastrous series for New Zealand in Australia, took a back seat in India's first innings when Jamieson and Tim Southee took four wickets apiece to roll the tourists for 165.
But he came to the fore on day three, first with the bat and then with figures of three for 27 as India, the world's top-ranked side, looked lost in New Zealand conditions.
"I'm just happy to be back out there, it's been a little frustrating on the sidelines," the left-armer said.
"If we can get a couple more early in the first hour (on Monday) or so, we should be in a good position."
Boult credited changing the angle of delivery as a key weapon to upset the Indian batsmen.
"We know it's a very good wicket generally and day three and four is the best time to bat. If I do my job, we won't be chasing too much."
Ashwin believed the pitch would give more in the final two days, bringing an emphasis on his spin if India can set a strong enough target.
"The Test match cricket, it begins now," he said.
"If you can get as close to the first innings score as they did then we might present ourselves an opportunity to do something, but that's very far away in the game and we have to take it every ball at a time because there is enough in the surface."
India lost Prithvi Shaw early in their second innings to give Boult his first wicket and he claimed Cheteshwar Pujara with the last ball before tea.
Mayank Agarwal went for 58 soon after the resumption and Boult removed Virat Kohli to have India at 113 for four. India had their opportunities to clip New Zealand's wings. They opened the day with two wickets in the first three overs to have New Zealand 225 for seven, just 60 runs ahead. But lusty hitting from Jamieson with an entertaining 44 off 45 balls and Boult adding 38 off 24, along with a more sedate 43 from Colin de Grandhomme saw New Zealand's last three wickets add 123 runs.
Ishant Sharma was the most successful of the Indian bowlers with five for 68 while Ashwin took three for 99.
=========================================== Scoreboard =========================================== India 1st innings 165 (A. Rahane 46; K. Jamieson 4-39, T. Southee 4-49) =========================================== New Zealand 1st innings (overnight 216-5) =========================================== T. Latham c Pant b Sharma 11 T. Blundell b Sharma 30 K. Williamson c sub (Jadeja) b Shami 89 R. Taylor c Pujara b Sharma 44 H. Nicholls c Kohli b Ashwin 17 B. Watling c Pant b Bumrah 14 C. de Grandhomme c Pant b Ashwin 43 T. Southee c Shami b Sharma 6 K. Jamieson c Vihari b Ashwin 44 A. Patel not out 4 T. Boult c Pant b Sharma 38 Extras: (b1, lb1, w6) 8 =========================================== Total: (all out, 100.2 overs) 348 ===========================================
Fall of wickets: 1-26 (Latham), 2-73 (Blundell), 3-166 (Taylor), 4-185 (Williamson), 5-207 (Nicholls), 6-216 (Watling), 7-225 (Southee), 8-296 (Jamieson), 9-310 (de Grandhomme), 10-348 (Boult)
Bowling: Bumrah 26-5-88-1 (6w), Sharma 22.2-6-68-5, Shami 23-2-91-1, Ashwin 29-1-99-3.
=========================================== India 2nd innings =========================================== P. Shaw c Latham b Boult 14 M. Agarwal c Watling b Southee 58 C. Pujara b Boult 11 V. Kohli c Watling b Boult 19 A. Rahane not out 25 H. Vihari not out 15 Extras: (w2) 2 =========================================== Total: (4 wkts; 65 overs) 144 ===========================================
Fall of wickets: 1-27 (Shaw), 2-78 (Pujara), 3-96 (Agarwal), 4-113 (Kohli)
Bowling: Southee 15-5-41-1, Boult 16-6-27-3, de Grandhomme 14-5-25-0, Jamieson 17-7-33-0 (2w), Patel 3-0-18-0
Toss: New Zealand Umpires: Richard Kettleborough (ENG), Aleem Dar (PAK) TV Umpire: Michael Gough (ENG)
Match referee: Ranjan Madugalle (SRI).