The Indian leg-spinner claimed 4 for 55 as the Kiwis, needing 384 to make India bat a second time, collapsed to 65 for 4 before a partial recovery carried them to 211 for 7 by close of the third day's play.
Nathan Astle held the tottering innings together with an unbeaten 68, adding 70 for the fifth wicket with his captain Stephen Fleming, and 45 for the unfinished seventh with Chris Cairns.
Astle and Cairns (18 not out) resume today with their side still needing 173 runs to save the follow-on and four wickets in hand. It is not an impossible task on the slow turner on which India declared in their first innings at 583 for 7. But Kumble, whose 10-wicket match haul won India the second Test at Kanpur last week, stands in New Zealand's way as they bid to wriggle out with a draw.
In Bloemfontein, Jacques Kallis took three wickets in 10 balls to steer South Africa towards a big Test victory over Zimbabwe, who were struggling on 123 for 5 in their second innings at the end of the third day, needing another 102 to avoid an innings defeat.
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