Asked to pull off a world-record chase when Australia declared their second innings closed at 8-270, India scored at a rapid rate on a sun-baked Saturday afternoon at The Oval to set up a big final day.
Kohli will start Sunday not out on 44, with Ajinkya Rahane alongside him on 20 and 280 more needed to pull off the highest successful chase in Test history and grab the global crown.
At one stage, India threatened to take the upper hand into day five, when Rohit Sharma and Cheteshwar Pujara were rattling along after tea to take the score to 1-92.
However back-to-back wickets for Nathan Lyon and Pat Cummins to remove the pair inched Australia back in front, knowing that early wickets on Sunday will expose India’s lower order.
The day was not without drama, with Cameron Green claiming a brilliant diving catch low to his left at gully off Scott Boland to remove opener Shubman Gill for 18.
Gill remained at the wicket and the decision was reviewed to the third umpire, before Richard Kettleborough determined Green had his hand under the ball and gave it out.
Sharma was visibly frustrated by the call, with the India captain appearing to believe the ball had brushed the grass after Green had gained control with his weaker left hand.
It came after Boland (1-38) was handed the new ball ahead of Mitchell Starc, in the Victorian’s pursuit of selection for the first Ashes Test.
It marked only the second time Starc has not opened, and instead bowled first-change for Australia, since 2015.
Starc was then Australia’s most expensive with the ball, going for 0-45 from his seven overs.
He was hooked for six by Sharma off the second ball he bowled, and has gone for eight boundaries in total.
Starc was not the only one to cop treatment off Sharma, who made 34 of his 43 runs in boundaries before Lyon got the crucial breakthrough.
Brought into the attack, the offspinner had one go straight on and trap the sweeping Sharma on the pads in front of the stumps.
Australia looked to have one hand on the trophy in the next over when Cummins had Pujara caught behind trying to uppercut him.
But, in worrying signs for Cummins’ men, India’s master batsman Kohli looked in good touch, using his feet well to Lyon and punching the ball down the ground off the quicks, amassing seven crisp boundaries.
Earlier, Alex Carey’s 61 gave Australia the chance to set a target of well above 400 after they lost Marnus Labuschagne for 41 in the third over of the day.
Supported by Green (25) and Starc (41), Carey put away the reverse sweep and batted patiently on a wicket with plenty of uneven bounce on Saturday.
After being dismissed on the shot four times this year against India, he did not play it once and instead nurdled Jadeja (3-58) around through his 105-ball knock.