This was just seventh Test match victory in England for the visitors
India beat England by 203 runs in the third Test at Trent Bridge on Wednesday to cut the home side's lead in the five-match series to 2-1.
England, set a mammoth 521 to win, were bowled out for 317.
India needed just 10 minutes to wrap up victory with Wednesday's 17th ball after England resumed the fifth day on 311 for nine, the match ending when Ravichandran Ashwin had England No 11 James Anderson caught by Ajinkya Rahane at slip — the first wicket for an India spinner in this fixture.
This was just India's seventh Test match victory in England.
It was also a personal triumph for India captain and man-of-the-match Virat Kohli who made 103 in his side's second innings 352 for seven declared following the star batsman's first-innings 97.
England, whose captain Joe Root won the toss and fielded, slumped to 161 all out in their first innings when they were punished by Hardik Pandya's five for 28 — the medium-pacer's maiden five-wicket haul in Test cricket.
They suffered another top-order slump in a second innings where they had to make cricket history to pull off an improbable win — no side have made more in the fourth innings to win a Test than the West Indies' 418 for seven against Australia at St John's, Antigua in 2003.
England were in dire trouble at 62 for four before a stand of 169 between Jos Buttler, whose 106 was his maiden Test century, and Ben Stokes (62) kept India at bay.
But the recalled India fast bowler Jasprit Bumrah then took three wickets for eight runs in five balls on his way to innings figures of five for 85 in 29 overs to leave England on the brink of defeat.
This win kept alive India's hopes of becoming just the second team, after a Don Bradman inspired Australia in 1936/37, to win a five-match Test series from 2-0 down.
It also represented a remarkable turnaround following their innings and 159-run loss in the second Test at Lord's.
"The victory was much needed at this stage of the series and to have a performance like that, which was so clinical in all three departments, is something that was pleasing to me as a captain," said Kohli at the presentation ceremony. "The players are really proud of what they've done in this Test too. Everyone is taking responsibility at the right time.”
Root admitted England's first-innings batting had let them down.
"It's fair to say that we very much under-performed in that innings," he said. "You look at the second innings and the partnership between Buttler and Stokes and that's a real lesson to our side in how to play Test match cricket.
He further stated: "Not the fact that they scored at a slower rate or looked to be more defensively minded, but the way they adapted to the situation and were very clear about how they were going to score their runs," Root added.
The fourth Test at Southampton starts on August 30.