New Zealand eye history after Santner's 7-53 in India Test
A stunned India fell to 156 all out in their reply, gifting the visitors a first-innings lead of 103 on a turning Pune pitch.
New Zealand, who have never won a Test series in India and lead the three-match series 1-0, were 198-5 at stumps in their second knock, a lead of 301, after skipper Tom Latham hit 86.
Wicketkeeper Tom Blundell, on 30, and Glenn Phillips, on nine, were batting at the close of play.
"Still a job to do with the bat," bowling hero Santner said.
"Obviously the more runs we get now, it makes our job with the ball a little bit easier."
Spinner Washington Sundar grabbed four wickets to take his match tally to 11 after returning figures of 7-59 on day one.
Sundar trapped Devon Conway lbw for 17 in New Zealand's second knock before senior spinner Ravichandran Ashwin sent back Will Young for 23 before tea.
Sundar also bowled Rachin Ravindra for nine as the left-hander went back for a cut only to miss and the skidding ball hit the off-stump. Daryl Mitchell fell for 18.
Latham reached his fifty and kept firm in a 60-run partnership with Blundell until his departure when Sundar trapped the left-hander lbw.
"It is satisfying," Santner said of the visiting batsmen taking on the Indian spinners in favourable conditions.
"We know how challenging it can be in these conditions. Seen India squeeze a lot of teams for a long time and I guess it was pleasing with the bat with guys trying to fire some shots and play some sweeps and take on the bowlers."
Kohli shock
Santner sparkled for New Zealand with his left-arm spin as he claimed his first five-wicket haul in 29 Test appearances to trump India at their own game of spin.
Ravindra Jadeja top-scored with 38 but India's batting fell apart, with the hosts bowled out in the second session in just 45.3 overs.
Overnight batsmen Yashasvi Jaiswal and Shubman Gill started cautiously before Santner struck in the 11th over of the day.
He trapped Gill lbw for 30 with a slider to the right-hander.
The home fans were then stunned into silence when Santner took the prized wicket of Kohli, bowled on a low full toss for one.
Santner admitted it was a "shock" to get the star batsman out in that manner.
Off-spinner Phillips struck on his fourth ball to get the left-handed Jaiswal caught at slip for 30 to leave the hosts on 70-4 and then 83-5 when he bowled Rishabh Pant for 18.
First-match centurion Sarfaraz Khan managed 11 before Santner got him caught at mid-on.
"I thought the pace that Santner bowled today was spot on for this surface," India bowling coach Morne Morkel said.
"You want to be playing cricket on the front foot in these conditions, but we couldn't find that rhythm and momentum today."
India resumed the day on 16-1 and their batting collapse was the second in the series after they fell to 46 all out in the first Test in Bengaluru for their lowest-ever home total.
Last week's win by New Zealand was their first Test victory on Indian soil since 1988.