England were 114 for two in their second innings at stumps on the third day, a lead of 154 runs, having already won this five-match series at 3-1 up.
Now Cook, England's all-time leading Test run-scorer and captain Joe Root (29 not out), his successor as skipper, will look to pile on the agony for India when they resume their unbroken third-wicket partnership of 52 on Monday.
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India's Ravindra Jadeja continued an excellent return to Test cricket by starring with both bat and ball.
Jadeja, playing his first Test of the series after being recalled in place of the injured Ravichandran Ashwin, added to his haul of four first-innings wickets by top-scoring with 86 not out in an India first-innings total of 292.
England, who led by 40 runs on first innings, have long struggled to find an opening partner for Cook, with 12 batsmen given a shot at the job since Andrew Strauss retired six years ago.
And ahead of their upcoming our of Sri Lanka, England may now be looking for two openers rather than one after Keaton Jennings completed a miserable series by being bowled for 10 playing no shot to Mohammed Shami on Sunday.
It was the second time in as many Test that Jennings had been dismissed leaving the ball and meant he ended the series with just 163 runs at a meagre average of 18.
An unconcerned Cook, who has already batted for more than three hours this innings, went 26 balls without scoring but seized on an over-pitched ball to drive Shami down the ground for four.
Moeen Ali, batting at number three -- another of England's problem positions -- failed to demonstrate similar patience.
The third member of England's all left-handed top three, Moeen drove loosely at Ishant Sharma on 14 only for normally reliable second slip KL Rahul to drop the head-high chance.
Fortunately for India, Moeen was bowled by Jadeja for 20 trying to drive the left-arm spinner out of the rough.
But Root, batting in his favoured position of number four, showed his class with a straight-driven four off Shami to take the score to 100 for two.
A frustrating day for England spearhead James Anderson saw him continue to remain two shy of Australia great Glenn McGrath's record of 563 Test wickets -- the most taken by a fast bowler.
Before play started, the International Cricket Council announced Anderson had been fined 15 percent of his match fee for "speaking in an aggressive manner" to umpire Kumar Dharmasena on Saturday after an lbw review against India captain Virat Kohli went the way of the star batsman.
Kohli eventually fell to Ben Stokes for 49, strengthening an England grip on the game established earlier Saturday by Jos Buttler making 89 on his 28th birthday.
India resumed in trouble at 174 for six, with Vihari 25 not out and Jadeja unbeaten on eight.
Vihari, who came through a couple of close lbw incidents involving Broad when on nought, was increasingly assured as he completed a fifty in 104 balls, including six fours and a six.
But a seventh-wicket partnership of 77 finished when off-spinner Moeen, whose hat-trick ended last year's Oval Test against South Africa, had Vihari caught off a thin edge by wicket-keeper Jonny Bairstow for 56.
Jadeja completed a 113-ball fifty, including seven fours and celebrated the landmark with a trademark twirl of the bat.
Silly-point Jennings then snatched at a bat/pad chance when last man Jasprit Bumrah went forward first ball to leg-spinner Adil Rashid.
Jadeja, on 56 at the time, cashed in on Bumrah's reprieve.
England took the new ball only for Jadeja to drive Anderson straight back over his head for six and hook Broad for four down to long leg.
But Jadeja, who made his highest Test score of 90 against England at Mohali two years ago, was left short of a hundred when Bumrah was run out as he tried to keep the strike.
First Published: September 10, 2018, 12:32 PM IST