England were 209-3 in reply to India's first-innings 471 at stumps, still a deficit of 262 runs, with Pope exactly 100 not out after being dropped on 60.
But they had been 4-1 when Pope came to the crease after a superb Bumrah delivery had Zak Crawley caught by Karun Nair off just the sixth ball of the innings.
And after Surrey right-hander Pope reached his century, a gripping day of Test cricket ended with late drama involving two England batsmen on their Yorkshire home ground.
Joe Root fell to Bumrah for 28 when he carelessly steered him straight to first slip.
And, in the last over of the day, it looked as if the fast bowler had dismissed Harry Brook for a duck when the rising star top-edged a needlessly aggressive pull and was well caught by Mohammed Siraj at midwicket.
But the third umpire confirmed a no-ball to spare Brook's blushes.
Bumrah ended the day with 3-48 from 12 overs, having taken all the England wickets to fall so far in the first of this five-Test series.
Pope came into this match following his 171 in last month's one-off Test against Zimbabwe.
But he was still under scrutiny, given his average in 13 previous Tests against India was a modest 24.60 - a figure inflated by his magnificent 196 at Hyderabad last year.
Pope, however, has long enjoyed the confidence of England captain Ben Stokes and coach Brendon McCullum.
"He probably couldn't walk out in tougher conditions, Jasprit running down the hill with the lights on," England opener Ben Duckett, who made 62 on Saturday and put on 122 for the second wicket with Pope, told reporters after the close.
"There's no better feeling than that, scoring a hundred against that attack after coming out at 4-1.
"You can see that in the way he celebrated, but it didn't just mean a lot to him, it meant a huge amount in the dressing room as well."
Bumrah could have had even better figures but for dropped catches with the usually reliable Ravindra Jadeja putting down Duckett at backward point on 15.
The fast bowler did dismiss Duckett via an inside-edged a drive onto the stumps.
But he had another chance missed off his bowling when Pope, repeating the steer shot with which he reached him fifty, was put down by a diving Yashasvi Jaiswal in the cordon.
'Best bowler in the world'
Duckett, who hailed Bumrah as the "best bowler in the world", added: "It is hard as an opener as he bowls so many different balls."
Pope went to 95 with a square-cut boundary off Shardul Thakur before an inside-edged single off Bumrah took him to a 125-ball century including 13 fours.
India had threatened a total in excess of 500 earlier Saturday after Pant (134) joined opener Jaiswal and captain Shubman Gill as one of three batsmen to make hundreds in the innings.
But they lost lose their last seven wickets for 41 runs, with all-rounder Stokes taking an economical 4-66 in 20 overs, while fast bowler Josh Tongue wrapped up the innings on his way to 4-86.
India resumed on their overnight 359-3, with Gill 127 not out in his first innings since succeeding the retired Rohit Sharma as skipper, and Pant unbeaten on 65..
Pant soon went to a hundred in stunning style, the wicketkeeper charging at Shoaib Bashir before hoisting the off-spinner high over deep midwicket for six despite one hand coming off the bat.
The 27-year-old then celebrated his century with an acrobatic somersault.
Pant's seventh hundred in 44 Tests, and fourth against England, came off 146 balls including 10 fours and four sixes in a typically aggressive innings.
The pair were in complete command until Gill's mistimed pick-up shot off Bashir flew straight to deep square leg as the 25-year-old skipper fell for 147, having faced 227 balls including 19 fours and one six.
Gill's exit ended a partnership of 209 in 49 overs with Pant that had taken India to 430-4 and opened the door for a flurry of wickets to give England hope.
Pant, having played a succession of extraordinary strokes, was lbw offering no shot to Tongue before the paceman polished off the tail.