Teenager Haseeb Hameed defended dourly to remain not out on 12 (1x4) from 84 balls while skipper Alastair Cook was unconquered on 28 from 84 balls with a couple of fours.
They did well to keep the five-man Indian attack at bay in front of a 20,000-plus crowd, the best turn-out of the match so far.
England now need 365 runs from four sessions as the task looks extremely difficult with only four teams to have made more than 400 in the fourth innings to win a Test.
India bowled in good areas but surprisingly the fourth day wicket which was perceived to be a rank-turner held firm as the English openers played each ball to its merit in their fight for survival.
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On a hat-trick, Ashwin was brought in the fifth over but Cook played him without any fuss, while Hameed held firm as England managed to keep themselves unscathed in the second session.
Hameed who was run out in the first innings, was very solid in defence as India looked for something special to dismiss the 19-year-old.
Kohli batted with his class and majesty stroking out shots all around the park in his knock from 109 balls, studded with four boundaries but before he could notch yet another century Ben Stokes' one-handed catch at the slip ended his stay.
Looking effortless all around the park, Kohli went for an expansive powerful drive throwing his bat wide of his body as the ball looped high in the air and looked as if it went past the giant frame of Stokes at the slip.
With Kohli showing some frustrated emotion going back, Stokes remained non-chalant but by then the lead had surpassed 350 and Jadeja started playing freely to step up the run-rate.
(REOPENS DEL 47)
But Mohammad Shami slamed a 22-ball 19 studded with two sixes and one four as he along with Jayant Yadav (27 not out; 4x4) added 42 runs from 58 balls in an extended lunch session.
Earlier a brave Stuart Board led England's fightback as he claimed with 4/33, while legspinner Adil Rashid returned with 4/82 but they struggled to dismiss the tail with some counterattack by the duo of Shami and Jayant.
Defeat looking imminent, the embattled England found some solace in Stuart Board's incisive bowling after India resumed in the penultimate morning on 98/3.
The tall pacer defied a strained tendon in his right forefoot and bowled a fine spell of 8-0-27-2, dismissing the overnight batsman Ajinkya Rahane (26) and Ravichandran Ashwin (7).
Broad may have broken the stand at 87, but the sharp bounce and the crack was a welcome for sign for Indian bowlers and concern for the English batsmen who would bat last on a worn-out Dr YSR Reddy ACA-VDCA Cricket Stadium pitch.
Having survived on six after a successful referral, Ashwin did not last long and became Broad's fourth victim edging one behind the stumps.