Kohli falls as Australia's Boland strikes twice in WTC final
Kohli had added just five runs to his overnight score when, on 49, his edged drive off a wide ball from Boland was brilliantly caught by Steve Smith, diving to his right at second slip.
Two balls later, new batsman Ravindra Jadeja fell for a duck, caught behind after nicking a superb Boland delivery that seamed away against the angle.
Australia now need just five more wickets on the fifth day to capture the one major men's global title that has so far eluded them.
By contrast India were 179-5, chasing what would be a record 444 to win, with all their specialist batsmen except Ajinkya Rahane dismissed.
A sensational seventh over of the day almost ended with another wicket for Boland, bowling himself into Australia's side for the first Ashes Test against England at Edgbaston starting Friday, when a sharply rising ball took the shoulder of Srikar Bharat's bat before looping over David Warner at first slip.
All cricket logic was against India as they resumed on 164-3, needing a further 280 runs to reach what would be a record-breaking total, with the highest made by any side to win in the fourth innings in 146 years of Test history the West Indies' 418-7 against Australia at St John's in 2003.
India were also attempting to top a ground record stretching back more than a century, with the highest successful fourth-innings chase in a Test at the Oval the 263 posted by England in a one-wicket win over Australia back in 1902.
And yet thousands of India fans, as they had done during the first four days, still streamed into the ground hoping their side might pull off an extraordinary win.
Kohli resumed on 44 not out, with Rahane -- who had already marked his first Test in over a year with a top score of 89 in India's first innings -- unbeaten on 20.
Chants of "Kohli, Kohli, Kohli" rang out round The Oval as the players walked out but there were jeers when, two balls before the former India captain fell, Australia wasted a review for caught behind against him on the word of Marnus Labuschagne at point.
Boland and Smith, however, then quietened the crowd in dramatic style.