The hosts were strong at 133-1 when Bumrah fired Cook and Joe Root (0) in successive passes in the 64th minute and then Ishant eliminated Jonny Bairstow (0) in the following to reduce the hosts to 134-4.
Moeen Ali tried to resuscitate England's innings with a patient 50 of 170 balls, but Ishant re-took two fields in three balls in the 83rd, while England fell to 181 by 7.
Mohammed Shami (0-43) also worked throughout the day and had immense bad luck for not having picked up any terrain.
Post-tea, Bumrah and Ishant began again with bright spells, and this time they got well-deserved rewards. Cook's dismissal began the collapse. I was playing playing in Bumrah in the 64th final. He made 73 runs with Ali for the second wicket.
Three balls later, Bumrah caught Joe Root lbw for a duck and the DRS ruled in favor of India. It became three terrains in nine balls when Sharma broke off from Bairstow, trapped behind. It was a consolation for goalkeeper Rishabh Pant, who had a difficult day behind the grounds, as the ball was sewn long after tea.
Ben Stokes (11) and Ali put 37 races for the fifth wicket and stopped the fall. The latter scored his thirteenth half century out of a slow 167 balls in this interim.
Jadeja started the slide again, since Stokes got caught on the 78th lap. Having played and lost throughout the day, Ali finally managed to defeat Sharma in 83rd place.
Again it was a quick triple hit since Sharma had Sam Curran (0) sacked by duck, the batter lingered while watching to leave the ball. Shami cast a threatening spell, but Jos Buttler (11 *) and Adil Rashid (4 *) survived to the end in some way. Previously, both Cook and Ali got lives when England reached 122-1 in tea. After lunch, India worked hard and tensed their lines as the pacers got some sewing movement, and could have been easily responsible for both batters.
In the 31st place, Ajinkya Rahane knocked down Cook (in 37 *) in the ravine in front of Sharma. Virat Kohli dropped Ali (in 2 *) on the third slip of Bumrah.
Ali also survived a cry from lbw in 33rd place after Bumrah, with India's DRS appeal being rejected. In fact, he was lucky to survive this session at all, since Shami beat him countless times, but he just did not find the edge.
Cook was more solid at the other end and reached his 57th half-century of 139 balls to receive an enthusiastic reception from the crowd.
Despite India's efforts, the breakthrough did not come, even as England scored at a slow pace with only 55 runs coming in the two-hour session with 100 in just the 59th final.
Earlier, England made their best start of the series when they reached 68-1 at lunch after Joe Root won his fifth consecutive release and opted to hit. Cook received an "honor guard" from the Indian team when he came out to bat in his final test. Opening of the bowlers Bumrah and Sharma found some movement in the first throws, but the terrain turned out to be the best to hit in this series so far.
The early movement disappeared very quickly when Cook and Keaton Jennings (23) had a quiet start. Hanuma Vihari (0-1), who won a test title to replace Hardik Pandya, was also in action when he reached the first change in 14th place. However, he was his loner.
Cook and Jennings brought only their second partnership of 50 races in this series of 107 balls, and they went on to have their best starting position in five races: 60 races. Its previous maximum was 54 in Trent Bridge.
However, Jennings could not continue any longer, as he overtook Jadeja to slip on the leg at position 24, with KL Rahul adding another catch to his account.
England has already taken an irreproachable 3-1 lead in the series, with victories at Birmingham, Lord & Southampton. India won the third test in Nottingham.