Only two teams have made more than 215 in the fourth innings of a Test to beat India -- Australia with 342-8 at Perth in 1977/78 -- and West Indies, who made 276-5 at Delhi in 1987/88.
The match was still somewhat in the balance when India captain Virat Kohli was out for 44 before lunch, with the tourists then 312-6, a lead of 213 runs.
But Pant, sensibly playing himself in rather than recklessly giving his wicket away as he'd done so often this series and Thakur, fit following a hamstring injury that had kept him out since the drawn first Test in Nottingham, were in fine form.
They took the game away from England's tiring pace attack on a good pitch, with a sunny day ideal for batting.
Thakur carried on from his first-innings 57 by driving off-spinner Moeen Ali through the covers and Craig Overton down the ground for two well-struck fours. Meanwhile, left-hander Pant lofted Moeen back over his head.
Even James Anderson, Test cricket's all-time most successful fast bowler, was made to look every one of his 39 years when Thakur hoisted him through deep midwicket.
And such was Thakur's command, with the ball no longer deviating in the air or off the pitch for England's quicks, he had no trouble in spotting a deliberately slower delivery from seamer Ollie Robinson and driving it high over long-on for six.
Thakur's third fifty in four Tests came in just 65 balls, including five fours. England captain Joe Root, a part-time off-spinner, broke the partnership when he Thakur edging a drive to Overton at slip.
Pant, a delivery after completing a 105-ball half-century, was caught and bowled by Moeen off a hard-hit drive.
But the runs kept coming, with Umesh Yadav hoisting Moeen for six while fellow tailender Jasprit Bumrah thumped Chris Woakes wide of mid-off for four -- even more worryingly for England, key batsman Root was left hobbling after trying to intercept the ball.
India resumed on 270-3, already 171 runs ahead.
Rohit Sharma had led the way with 127, his first overseas Test century, and together with Cheteshwar Pujara (61) he frustrated England during a second-wicket partnership of 153 -- although it would have been a different story had not Sharma been missed twice in the slips.
But Warwickshire all-rounder Woakes, who had already marked his first Test in more than a year with first-innings contributions of 4-55 and 50, reduced India to 296-5 with a burst of 2-0 in nine balls.
The paceman struck with just his second ball of the day when he had left-hander Jadeja lbw for 17 and followed up by having Ajinkya Rahane lbw for a duck as the struggling India vice-captain aimed across the line.
Star batsman Kohli was in fine touch until he edged Moeen to Overton at slip for 44 -- his 21st successive Test innings without a hundred.
India head coach Ravi Shastri was unable to watch proceedings at the Oval, having been forced to remain in the team hotel after testing positive for Covid-19 late Saturday.