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New Zealand were 125 for four as they entered the third session of play.
The visitors, who resumed their second innings at their overnight score of 4 for one, lost four more wickets in the post-tea session.
But the last pair of debutant Rachin Ravindra (18) and Ajaz Patel (2) held on for 52 deliveries to deny the hosts victory.
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Brief Scores India: 345 and 234/7 declared.
New Zealand: 269 and 165 for 9 in 98 overs (Tom Latham 52, William Somerville 36; Ravichandran Ashwin 3/35, Ravindra Jadeja 4/40).