Pakistan, forced to follow-on 268 runs behind after being dismissed for 407 in their first innings, were reduced to 207 for nine by stumps in their second knock despite an unbeaten 107 from Yousuf Youhana on Wednesday.
The tourists will return on the final day on Thursday to complete an unprecedented win after their previous 20 Tests on Pakistani soil ended in five defeats and 15 draws.
India claimed 13 wickets on the fourth day after just 11 had fallen on the first three days on the barren wicket slammed by critics as being too much in favour of batsmen.
The match would have ended on Wednesday itself, but for Youhana's gallant century and his 70-run stand for the ninth wicket with Shoaib Akhtar after Pakistan were reduced to 136 for eight.
Akhtar defended stoutly for an hour to make four before he became Kumble's sixth victim during the extra 30 minutes claimed by the Indians to force a result by stumps.
Kumble, who took all 10 Pakistani wickets in an innings in New Delhi in 1999, returned with his 24th five-wicket haul to take his overall tally to 390 from 82 matches.
Left-arm seamer Irfan Pathan followed a career-best 4-100 in the first innings with one for 22 in the second.
The Indians overcame the temporary loss of strike bowler Zaheer Khan due to a leg injury to shatter the Pakistani batting.
Pakistan, who began the day at 364 for six in their first innings in reply to India's 675 for five, lost their last four wickets for 43 runs to be bowled out for 407 after the first drinks break.
Only a last-wicket stand of 36 between Mohammad Sami (15) and Shabbir Ahmed (19 not out) boosted the total.
Pathan began the destruction by removing Abdul Razzaq with the first delivery of the morning, a rising ball that found the edge to wicketkeeper Parthiv Patel.
When Pakistan batted a second time, they collapsed to 44 for three after skipper Inzamam-ul-Haq was run out without scoring.
Inzamam, victim of a doubtful decision in the first innings after making 77, walked in to loud applause from some 5,000 fans on his home ground - only to be return soon after.
He responded to a call from Yasir Hameed for a sharp single, but failed to make the striker's end as Yuvraj swooped on the ball at mid-wicket and threw down the stumps with a direct throw.
Yuvraj was at it again when, bowling left-arm spin, he claimed his first Test wicket as Hameed's pull shot was deflected off the close-in fielder's body to Virender Sehwag at fine-leg.
Soon after tea, Kumble removed Abdul Razzaq for 22, superbly caught at short-leg by Akash Chopra as Pakistan slipped to 106 for five.
Moin Khan was trapped leg-before by Pathan, before Kumble dismissed both Mohammad Sami and Saqlain Mushtaq in similar fashion.
But Youhana, who hit 15 boundaries and two sixes in his 11th Test hundred, delayed the inevitable till Thursday.