Until Ravindra Jadeja had Cook lbw pushing forward in the last over, realistic hope endured that England might yet salvage a stalemate in Vizag.
They faced an unenviable task after taking seven India wickets in an extended morning session on day four to leave themselves facing a national-record 405-run chase to win - or alternatively bat out a minimum 150 overs to keep the series level at 0-0 with three to play.
It would be a monumental effort to achieve either, but Cook and Haseeb Hameed sowed belief in a painstaking opening stand of 75 in 50 overs to help England reach stumps on 87-2.
India had stumbled from 98-3 to a lunchtime 204 all out, to Stuart Broad (4-33) and Adil Rashid (4-82), with Mohammed Shami and Jayant Yadav’s last-wicket stand accounting for 42 of their runs.
England’s stoic openers were never going to threaten any speed records in reply on a pitch which had shown signs of deterioration as early as day two.
It played relatively few tricks at the start of the fourth innings, however - and Cook (54), with the slowest half-century of his career, and Hameed proved equal to anything India could muster until deep into the evening session when the surface began to offer sporadic turn and variable bounce for Ravi Ashwin and Jadeja.
It was not until the 38th over that England inched to 50, Cook and Hameed having registered just two boundaries each.
Cook upped the tempo marginally in the final session, during which he survived both India reviews for lbw in the space of six balls on 45 - Jadeja and then Ashwin the men disappointed by umpire’s call.
Yet it was a measure of the challenge still towering over England that their highest successful run chase in any climate is still their 332-7 in Melbourne 88 years go - while the top score by any tourists in India remains West Indies’ 276-5 to draw in Delhi in 1987.
England’s best was 241-5 in a Madras stalemate 52 years ago.
Hameed’s admirable three-hour vigil ended when he fell victim to an Ashwin off-break that barely bounced and sneaked under his bat to pin him in front lbw.
But Cook stayed put to reach his 50 from 172 balls in almost three-and-a-half hours, and seemed set to close out the day alongside Joe Root until Jadeja struck.
Broad had earlier defied the pain of the tendon strain in his right foot, diagnosed the previous evening, to double his wicket tally.
It was Rashid who got the prize wicket of Virat Kohli, 19 runs short of what would have been the India captain’s second century of the match.
But Broad began India’s relative struggles.
His dismissal of Ajinkya Rahane doubtless furrowed the brows of England’s batsmen amid the celebrations - because a short-of-a-length ball kicked alarmingly to take the edge for a routine slip catch.
Broad then made short work of Ashwin, caught behind via a faint outside-edge on an attempted push-drive, and Rashid soon got Wriddhiman Saha too - lbw to a googly.
Kohli departed to the leg-spinner, thanks to a memorable catch at slip as Ben Stokes dived to his right to hold on one-handed, and Jadeja’s plan to go after Rashid brought him a six over long-on before he holed out at deep midwicket.
Umesh Yadav fell for a duck to a sharp catch by Jonny Bairstow, trying to mow runs to leg off Rashid. But the 10th-wicket pair delayed lunch until the belated introduction of Moeen Ali resulted immediately in the stumping of Shami.