After electing to bat in the historic 500th Test, India were comfortably placed at 154 for riding on 112-run second wicket stand between Vijay (65) and Pujara (62).
However a disciplined bowling effort complemented by some inept shot selection by middle and lower order saw four wickets for 16 runs after the second new ball was taken.
Rohit Sharma (35) once again got a start only to throw it away at the fag end of the day. Ravichandran Ashwin was well set before being dismissed to a brilliant delivery from Trent Boult.
Ravindra Jadeja was batting on 11 at the end of the day with Umesh Yadav (8).
More From This Section
The Green Park track was not a rank turner on the first day itself but it did offer some assistance to the spinners.
The Kiwis fielded three spinners - Santner, offie Mark Craig and leg-break bowler Ish Sodhi, who bowled 59 out of the 90 overs.
Rohit (35) and Ashwin (40) combined for a fighting 52-run stand for the sixth wicket.
In an important phase in his Test career, Rohit was expected to fire and he showed some early promise before trying to chip one off Santner over mid-on but was holed out by Sodhi.
The other two Indian batsmen who played poor shots were skipper Virat Kohli (9) who fell for Neil Wagner's short ball mistiming a pull-shot.
It was left-arm spinner Santner, who removed prodigious Rahul in the morning session and later broke the second-wicket stand between Vijay and Pujara to open the window for his side.
Wagner bowled his heart out in the second session and was rewarded with the prized wicket of Indian captain Kohli.
Five minutes before the tea, Vijay tried to cut a shortish delivery from leg-break bowler Sodhi but managed a faint edge, taken by stumper BJ Watling.
Vijay was mostly patient from one end, treating the balls on merit.
After his Caribbean sojourn was marred by injury and the only Test match he got a chance to bat didn't go well, it was a good opportunity for the Tamil Nadu opener, who would rue the chance of having missed out on a big score after getting set.
Craig (1/59) though saw the back of Rahane (18 off 36) getting him caught by Tom Latham at forward short leg.
Spin all-rounder Ashwin turned to be a solid support for Rohit Sharma, who himself survived a confident leg before appeal from Craig.
Ashwin was comfortable playing both pacers and spinners, driving elegantly before being undone by Boult.
Each and every Kiwi bowler was among wickets, highlighting a team effort, led by Santner.
Vijay and Pujara continued in their nonchalant way in the post-lunch session, untroubled by the Kiwi bowlers.
Vijay guided Boult to third man with a late cut to touch 49 and survived a leg before appeal off Sodhi before reaching his fifty with a single off Boult in the next over, the 41st of the innings.
Vijay preferred to play his shots from the backfoot against both Sodhi or Boult while Pujara was fantastic with his nimble footwork.
With nothing working for the Kiwis, New Zealand captain Kane Williamson tried to attack more by employing two close-in fielders -- a slip and a forward short-leg man - when Santner operated.
Wagner did get a bit of reverse swing but the well-set Indian batsmen dealt with it without fuss.
Santner provided the breakthrough again but it was a soft dismissal as Pujara committed to a shot with the ball stopping on him offering a simple return catch.
Pujara's 62, his eighth Test half-century, came off 109 balls with eight shots to the fence.