Cook ended day four on 168 not out, making him the first batsman to make 100 in his first three Tests as captain.
An unbroken sixth-wicket stand of 141 with Matt Prior, who finished on 84, kept England's hopes alive as they ended the day on 340-5, despite a run of falling wickets earlier in the day.
And with Prior, the captain finally found the lasting support he so badly needed and deserved as his near eight-and-a-half-hour triumph of technique and determination on a surface increasingly favouring spin underpinned a stumps total of 340 for five which sneaked England into a lead of 10.
Cook's 21st Test hundred took him past his mentor, and England's current batting coach, Graham Gooch in the list of those to make the most for his country.
Cook and Nick Compton's century opening stand continued this morning, until the debutant faltered.
Compton had nonetheless eaten up 128 balls and lasted 45 overs for his 37 runs - a handy contribution in the context of the match - before he toppled over in defence against Zaheer Khan's left-arm inswing, and fell lbw.
Jonathan Trott then appeared to be getting himself in, after his first-innings duck, and could hardly be criticised for edging behind on the forward-defence when Pragyan Ojha turned one sharply from a perfect line and length.
Kevin Pietersen has long disputed he has a weakness to left-arm spin, but fell to that variety for the 25th time in his career when Ojha followed up yesterday's success against him.
Bowled middle stump playing inside one that turned then, this time Pietersen got so far across to sweep that he was bowled round his legs by a delivery that took the off bail.
Ian Bell was under extreme pressure after his rush of blood against Ojha, and the resulting embarrassment of a golden duck yesterday.
But he confidently drove his second ball from the same bowler through extra cover for four, and soon afterwards Cook completed his 181-ball century in more prosaic fashion with a push for two into the leg side off Yadav.
Bell went in early afternoon, the first of two lbws in two deliveries for Yadav - thanks to reverse swing, with a ball almost 80 overs old.
Both Bell and then Samit Patel could have just as easily survived, each time the ball shaping in towards the outer limit of leg stump, with a suspicion of inside edge on the second occasion.
Prior top-scored in an England first innings he himself described as a "shocker" last night, and it was soon clear he retained an appetite to try to put things right.
Like Cook, his second attempt was chanceless - although both could have had few complaints to go lbw to Ojha, the opener on 41 and Prior on 65.
If they had one minor piece of good fortune each, it was earned.
Cook barely put a foot wrong throughout, and never wavered in stamina-sapping heat and glare.
Prior, like his captain, employed impeccable shot selection - and when, inevitably, they were beaten by the spin of Ojha and the wicketless Ravichandran Ashwin they did not fret about what was still to come, and were therefore able to withstand it.
This was not mere survival either, and it was testament to England's resourcefulness that their two first-innings tormentors have each conceded more than 100 runs second time round.
Cook and Prior's combined efforts may yet prove in vain, but acknowledgement is due already for taking this previously one-sided match much deeper than so many so knowingly predicted after England's initial haplessness.
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