Tom Blundell was unbeaten on 30 and Glenn Phillips was on nine as the Black Caps, who won the series opener by eight wickets, extended their lead to 301 runs at the close on Friday after scoring 259 in the first innings.
"It's a great feeling. To put ourselves in a position to potentially win a Test here is pretty cool," Santner said.
Tom Latham's New Zealand could hand India their first homes series loss since 2012. (AP PHOTO)
"The way the guys did it with the bat there, it doesn't look easy and we knew getting partnerships was the key. We managed to do that today, which was outstanding."
Washington Sundar got rid of Devon Conway for 17, Rachin Ravindra for nine and Daryl Mitchell for 18 while Ravichandran Ashwin dismissed Will Young for 23, but the runs came quickly and too easily for New Zealand.
New skipper Tom Latham returned to form with a timely 86 but became Washington's fourth victim late in the afternoon after he put the tourists in a strong position to secure their maiden series victory in India.
India are seeking a 19th straight home series victory since a 2-1 defeat by England in 2012, but will need a battling effort to draw level with New Zealand ahead of the finale in Mumbai early next month.
Santner had earlier posted career-best figures of 7-53 while Phillips bagged 2-26 as the spinners continued to sparkle at the Maharashtra Cricket Association Stadium, where Washington took 7-59 on the first day.
The hosts resumed on 1-16 after losing captain Rohit Sharma for a nine-ball duck, and Shubman Gill was the first key wicket, trapped plumb in front by Santner for 30.
The left-armer struck again by bowling Virat Kohli for one with an inviting full toss that the batsman missed completely.
Yashasvi Jaiswal drove and swept the bowlers and appeared a lot more at ease, although the left-hander was lucky his slog sweep off Ajaz Patel fell in no man's land when he was on 16.
Virat Kohli walks off after being bowled by a Santner full toss in Pune. (AP PHOTO)
Mitchell pulled off a sharp catch at slip as Phillips dismissed Jaiswal for 30 and the tourists could have had another had the fielder held onto a tough chance when Santner forced an edge off Sarfaraz Khan.
Rishabh Pant was castled for 18 by Phillips after a loose shot and Santner had Sarfaraz caught for 11 to leave India on the ropes at 6-95, with both their batting heroes from the opening game back in the dugout.
Santner trapped Ashwin lbw for four and removed Ravindra Jadeja in similar fashion after the batsman made a defiant 38, before getting one to go through Akash Deep's defence.
He then snared Jasprit Bumrah for a duck to ensure New Zealand began their second innings with a lead of 103.
