Cook was still battling away unbeaten, on 46, as he so often has throughout his 161 Tests, in a total of 114-2 as England extended their overall lead to 154 after bowling India out for 292.
Home advantage on day three of this fifth Test would have been still more substantial without the revival engineered by India’s charismatic all-rounder Ravindra Jadeja (86 not out) and Hanuma Vihari (56).
The tourists resumed on 174-6 and appeared sure to trail significantly until the seventh-wicket pair took their stand to 77, and 132 were added in all for the last four to fall.
As Cook then set out to consolidate, he was cheered and clapped to the middle by his fourth standing ovation of the week here – before a handshake from India captain Virat Kohli as he prepared to take guard.
He soon lost his opening partner Keaton Jennings – in worrying circumstances as the man still expected to try to form the backbone of England innings into this winter and perhaps beyond shouldered arms to Mohammed Shami and lost his off-bail.
It was an unedifying end to Jennings’ largely fallow Test summer with the bat, especially after he had earlier dropped the catch at short-leg which reprieved India number 11 Jasprit Bumrah and cost England 32 runs.
Six days after his 'nothing-left-in-the-tank' declaration, Alastair Cook is still running, in better condition than at any time during the summer. On the cusp of another 50, what odds he keeps motoring to a farewell century?— Peter Jackson (@JackoRugby) September 9, 2018
Cook spent 26 balls stuck on 13 either side of tea.
But he uncovered occasional fluency, in an unbroken 50 stand with his captain Joe Root after Jadeja had spun one through the gate to bowl Moeen Ali – and there were no major alarms to the close.
As has been the curious case throughout this match so far, bat dominated ball before lunch – thanks to Jadeja and Vihari this time.
James Anderson, still three wickets short of overhauling Australia great Glenn McGrath’s world record of 563 wickets, drew a blank alongside Stuart Broad.
Debutant Vihari, who would have been lbw for nought to Broad the previous evening if England had gone to DRS, completed his half-century from 104 balls.
It took Jadeja, returning here for his first Test of the series, nine deliveries more and into the afternoon before he cut Moeen for his seventh boundary to reach the same milestone.
Vihari’s dismissal, caught-behind pushing up the line as Mooen’s arm ball drifted across him, was the only one before lunch.
Ishant Sharma went in near action replay to Moeen in early afternoon, and then Shami got greedy against Adil Rashid and holed out at long-on.
Jadeja would have been left stranded at 260 all out if Jennings had held a sharp catch off bat and pad when Rashid’s googly was too much for Bumrah.
But he did not – and Jadeja, who had celebrated his 50 with a trademark swish of his imaginary Rajasthan warrior’s sword – cashed in as the buccaneering presence in a last-wicket stand.
As Root’s field placings and bowling changes sent out mixed messages, Bumrah faced only 14 out of 54 deliveries – and Jadeja memorably smashed the second new ball straight back over Anderson’s head for six.
India’s fun, and England’s frustration, finally ended when Bumrah was run out chancing a tight single. Then, of course, it was over to Cook – one last time.
National selector Ed Smith is keen for the England set-up to make use of Alastair Cook‘s knowledge and experience after his retirement.
Former captain Cook, England’s all-time record run-scorer, announced earlier this week that he will retire from international cricket following the fifth Test against India.
When asked if he would like to involve Cook sometime in the future as a batting coach or in some other role with England, Smith told BBC Radio 5 Live’s Sportsweek programme: “I’m not sure about what he wants to do – he’s going to play for Essex for a while. He may just look at his options.
“I think it would be terrific for England cricket to tap into him as a resource.
“I’ve had the odd chat with him about bowlers and batters this summer, but next year maybe we’ll have something a bit more formal where he comes and really advises.
“He’s had so many games as captain, a record there, as a player he’s the record run-scorer in history, and he’s a calm, balanced and fair person, so I think we’ve got to try to tap into that knowledge and wisdom without a doubt.
“It doesn’t have to be necessarily that structured, but someone people know they can talk to as a trusted advisor.”
Buttler (89) marked his 28th birthday by top-scoring for the fourth time in England’s last five innings as he and Stuart Broad’s ninth-wicket stand of 98 underpinned a recovery from 198-7 at start of play to 332 all out.
Then in an impressive, occasionally bad-tempered but very skillful third spell of 6-2-8-2, James Anderson struck in successive overs to put himself just two wickets short of Glenn McGrath’s world-record 563 for any pace bowler.
As the old ball swung, Ben Stokes got in on the act with two late wickets – including prolific India captain Virat Kohli – in a stumps total of 174-6.
England had surprisingly dominated the morning – thanks to their driving force Buttler.
He received support from Adil Rashid and then especially Broad as England put themselves back on the front foot in pursuit of a 4-1 series scoreline.
Jasprit Bumrah pinned Rashid lbw, and Broad entered the equation on a still under-par 214-8.
England’s number 10 had 10 of a further 20 runs when Bumrah struck him a painful blow to his shoulder.
Far from losing his appetite for the battle, though, Broad’s concentration appeared redoubled.
India were unable to stop England’s charge for 20 overs until Broad tried to hit Ravindra Jadeja (4-79) over the top and fell to a brilliant catch by KL Rahul – chasing back from mid-on to dive and hold a skier.
There was to be no birthday hundred for Buttler, but he was not done with just yet.
Most Test wickets agains India:— BrokenStats (@Broken_Statz) September 8, 2018
107* James Anderson(51 inns)
105 M Muralitharan(32 inns)
94 Imran Khan(38 inns)
76 MD Marshall(30 inns)
69* Stuart Broad(35 inns)#INDvENG
He hit Bumrah for two sixes in three balls – into the crowd over long-on and then a hook towards the dressing-rooms – before he was last out, caught at slip trying to farm more runs off Jadeja.
By then, England’s last three wickets had put on 151 runs, and the onus was on India – all the more so after Broad had Shikhar Dhawan lbw.
Rahul responded, in company with the much more cautious Cheteshwar Pujara, in a second-wicket stand of 64 which ended when Sam Curran bowled the opener.
The all-rounder had his first Test wicket on his home ground, hinting at his habitual inswing but then getting the ball to leave Rahul off the pitch and take his off-bail.
Kohli survived thanks to a tight umpire’s call on 16 when Anderson thought he had him lbw – only for replays to vindicate Kumar Dharmasena’s on-field judgement of impact outside off-stump.
Anderson’s mood was lightened when Pujara edged behind, and then Ajinkya Rahane got a little more bat on the swinging ball and was caught at slip for a duck.
Debutant Hanuma Vihari made a nervy start, twice narrowly surviving Broad lbw appeals on nought, before keeping his captain company in a half-century stand.
But Kohli fell one short of his 50 when he edged a drive at Stokes to his opposite number at second slip, and left-hander Rishabh Pant was then caught at first in near mirror image.