Spinner Ravindra Jadeja claimed five wickets as India bowled out New Zealand for 174 on Sunday, leaving the hosts needing a tricky 147 to win the third Test.
New Zealand, who lead the three-match series 2-0, resumed on 171-9, losing Ajaz Patel for eight off Jadeja after seven minutes of play on day three at Mumbai's Wankhede Stadium.
The left-handed Ajaz was caught at deep mid-wicket off a slog sweep as left-arm spinner Jadeja ended with a match haul of 10 wickets. He also claimed five in New Zealand's first innings.
Fellow spinner Ravichandran Ashwin took three wickets.
The hosts are seeking to avoid being swept 3-0 after New Zealand won the first two Tests to register their first Test series triumph in India.
Will Young top-scored with 51 on Saturday on a track that has turned sharply since day one.
Shubman Gill hit 90 in India's 263 which gave them a slender 28-run first-innings lead in response to New Zealand's 235.