England, at lunch on the third day, were 124 for four in reply to India's record 664 - a deficit of 540 and still needing a further 341 to avoid the follow-on.
Leg-spinner Kumble, whose 110 not out on Friday was his maiden Test century and the first by an India batsman this series, removed both Alastair Cook and England captain Michael Vaughan shortly before lunch on his way to figures of two for 25 in 6.3 overs.
Cook, who'd been twice dropped off legside shots, ultimately saw one of his strengths become a weakness when he fell for 61.
The left-hander, trying to turn Kumble to the on-side, got a leading edge and was caught by Rudra Pratap Singh, running round from mid-off.
Kumble, in his 118th Test, then moved into joint third place on 563 wicket with retired Australia quick Glenn McGrath in the list of Test cricket's most succesful bowlers when he caught and bowled Vaughan for 11 off the last ball before lunch following the batsman's chipped drive.
Kevin Pietersen was four not out.
India, 1-0 up, had simply to avoid defeat in this match to win only their third Test series in England in 15 tours dating back to 1932.
England, who needed to win this match to preserve their six-year unbeaten run in home Test series, resumed on 24 for one.
Cook was 12 not out and nightwatchman James Anderson, fortunate to be still in after surviving a well-merited lbw appeal on nought, was unbeaten on five.
Cook had only added one to his overnight score when he was given the first of his two reprieves when leg-slip Dinesh Karthik, failed to hold the left-hander's glance off fast bowler Shanthakumaran Sreesanth with England 30 for one.
The Essex batsman had moved on to 28 when another leg-glance, off Kumble, was dropped by leg-slip Sourav Ganguly, a harder chance than the one Karthik had floored.
Kumble had a worrying moment when Anderson accidentally trod on his hand as he tried to field the ball off his own bowling.
Anderson, who'd hung around gamely for over an hour while adding 66 for the second wicket with Cook, was eventually out for 16 when left-arm swing bowler Singh had him plumb lbw on the backfoot. His exit left England 72 for two.
Cook square-cut Singh for four and subsequently drove him through extra-cover for another boundary that saw the 22-year-old to a 77-ball half-century - the 12th time in his 21 Tests he'd got to fifty.
India's total was their highest Test innings score against England, surpassing the 628 for eight declared they made at Headingley in 2002.