The Black Caps resumed at 198-5 on day three of the second Test and although they were dismissed inside the first hour of play on a turning pitch, India was left needing a miracle in Pune.
The highest successful Test chase for India at home was 387 against England at Chennai in 2008.
New Zealand leads the three-match series 1-0 and has never won a Test series in India.
India spinners Washington Sundar and Ravindra Jadeja shared seven wickets between them, with Jadeja claiming three of the five to fall on Saturday morning.
Glenn Phillips hit an unbeaten 48 after starting the day on nine.
Wicketkeeper Tom Blundell added 11 to his overnight 30 before being bowled by Jadeja.
Jadeja soon got Mitchell Santner for four and Ravichandran Ashwin took down Tim Southee for a duck.
The innings ended with the run-out of William O’Rourke, leaving Phillips stranded after a knock of four fours and two sixes.
Skipper Tom Latham top-scored with 86.
Left-arm spinner Santner put New Zealand firmly in the driving seat on day two with 7-53 to bowl out India for 156.
India are staring at their first home Test series loss since December 2012, when Alastair Cook’s England beat them 2-1 in a four-match series.
Mutib Khalid is a skilled content writer and digital marketer with a knack for crafting compelling narratives and optimizing digital strategies. Excel in creating engaging content that drives results and enhances online presence. Passionate about blending creativity with data-driven approaches, Mutib Khalid helps brands connect with their audience and achieve their goals.