Trent Boult has caused India all sorts of trouble late on day three, claiming three wickets as India limped to 144/4 at stumps.
India got off to a fast start in their second innings, before Boults got youngster Prithvi Shaw (14) caught behind.
He produced a stunning delivery to bowl Chesteshwar Pujara (11) as the Indian batsman looked to leave one that swung back in.
Tim Southee's toil was rewarded as he dismissed opener Mayank Agarwal for 58.
But the biggest wicket in world cricket was left for Boult, a short ball gloved down the leg side from Virat Kohli (19) sent the Indian captain on his way.
Ajinka Rahane (25) and Hanuma Vihari (15) are both not out overnight with lots of work to do on day four to get the visitors back in the test.
Earlier, Colin de Grandhomme and Kyle Jamieson put on a good partnership to guide the Blackcaps to a 183 run lead as they were bowled out by India at the end of the first session on day three.
A collapse looked likely when India made an early breakthrough on the third morning of the first cricket test in Wellington - nabbing two early wickets.
BJ Watling went first ball of the day for 14, while Tim Southee was dismissed a couple of overs later for six.
But de Grandhomme (43) and Jamieson (44) dug in and an entertaining 38 from no. 11 Trent Boult saw New Zealand finish on 348 in what is a more than handy lead.
The Blackcaps started day three with a 51-run first-innings lead at the Basin Reserve after bowling the visitors out on Saturday.
India had a good final session on day two, removing Ross Taylor, Kane Williamson and Henry Nicholls before stumps.
Captain Williamson batted magnificently before falling 11 runs shy of another century.
"I thought it was a great innings - a great partnership from him and Ross through the middle there," paceman Tim Southee said of Williamson's knock.
"We've got two guys that are very capable at the crease at the moment."
De Grandhome and Watling resumed on 216/5, with play getting underway at 11:30am.
On day two, Ishant Sharma took three wickets, snaring openers Tom Latham (11), Tom Blundell (30) and Taylor, while Ravi Ashwin (Nicholls) and Mohammad Shami (Williamson) took one a-piece.
India started the day 122/5, adding 43 runs to their total before being bowled out, with debutant Jamieson (4/39) and Southee (4/49) taking four wickets each.
Boult (1/57) also claimed a wicket, with the other coming by way of a run-out.