India's Bumrah produced a devastating spell in the morning on Friday, dismissing Ben Stokes, Joe Root and Chris Woakes before Archer struck with his third delivery to remove Yashasvi Jaiswal in his first Test for four years.
India ended the day on 3-145 in reply to England's 387, the game evenly poised with both sides bidding to go 2-1 up in the five-match series.
Earlier, Bumrah had reduced England to 7-271 in two fiery overs, bowling Stokes for 44 and Root for 104 before dismissing Woakes first ball.
Root, 99 not out overnight, had completed his century off the first ball of the day, flashing hard at Bumrah and leaping in the air as it flew past gully to the third man boundary.
It was his 37th Test hundred after another masterful display from England's highest Test run scorer.
Bumrah delivered another searing delivery that crashed into the top of Stokes' off stump before snaring Root with another peach of a ball that jagged back to remove his middle stump.
"The frustrating thing is I managed to get to 100 but that's when you should make it count," Root said. "Something you pride yourself on is turning it into a big score, turning the screw, so I'm a bit disappointed."
Woakes nicked a catch to stand-in wicketkeeper Dhruv Jurel to put Bumrah on a hat-trick, but Brydon Carse played a solid defensive stroke to deny him and then led the counter-attack with Jamie Smith, who reached his 50 just before lunch.
Smith's innings of 51 ended early in the afternoon session when he edged Mohammed Siraj to Jurel, ending a valuable eighth-wicket partnership of 84.
Archer was bowled by Bumrah, who finished with impressive figures of 5-74, but Carse reached his maiden test 50 before being bowled by Siraj for 56 to end the innings.
After a wayward first over by Woakes costing 13 runs, Archer took the ball.
His third delivery, timed at 90 miles per hour, clipped the edge of Jaiswal's bat and flew to second slip where Harry Brook pouched a sharp catch.
Archer wheeled away in delight before being mobbed by his teammates, the relief on his face evident after a succession of injury problems had threatened his career.
"The pure joy everyone has seeing him back in whites, everyone's genuine excitement to see him playing Test cricket again," Root said.
"Bowling 94mph, getting a wicket first over, that's the kind of player he is. He has the X-factor."
KL Rahul and Karun Nair defended staunchly in a second-wicket partnership of 61 and were starting to score more freely when Nair, on 40, edged Stokes to Root at first slip and he took an excellent low one-handed catch.
Nair reviewed the decision, which was confirmed, and Root moved top of the list of most Test catches by a non-wicketkeeper with 211 to his name.
India captain Shubman Gill arrived at the crease fresh from scoring 430 runs in the last Test at Edgbaston but he edged an outswinger to wicketkeeper Smith off Woakesto depart for 16.
Rahul, though, reached his half-century from 97 balls and was 53 not out at the close, with Rishabh Pant unbeaten on 19.
"It's quite evenly balanced but slightly ahead in our favour," said Bumrah.
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