Victory, achieved with more than a day to spare, left England an unassailable 3-1 up in the five-match series against India, the world's number one ranked Test side, ahead of Friday's finale at The Oval.
India, set 245 to win, were dismissed for 184 with recalled off-spinner Ali taking four for 71 in 26 overs on a wearing pitch, including the key wicket of star batsman Virat Kohli (58).
That gave Ali a match return of nine for 134 following his first-innings five for 63.
Unsurprisingly Ali, who also made 40 in England's first-innings 246, was named man-of-the-match.
It was another star turn at Hampshire's headquarters from Ali, who took six for 67 when England beat India by 266 runs in the corresponding Test at Southampton four years ago.
"Sometimes you can play so much that I felt it was not a bad thing to go back to county cricket," said Ali, playing his first Test of the season after being dropped by England following a difficult tour of Australia and New Zealand.
"When I was watching the guys at home I knew I really missed it."
Meanwhile elated England captain Joe Root insisted: "The strength of character has always been a massive strength of this team.
"But then the skill level to back it up has been phenomenal. I couldn't be more proud of the group," he added.
While India captain Kohli and his deputy Ajinkya Rahane (51) were sharing a century stand for the fourth wicket, the tourists had hope of chasing down their stiff target.
But after Ali dismissed both senior batsmen either side of tea, India lost four wickets for 13 runs.
Sam Curran, capping another fine all-round display, ended the match by having Ravichandran Ashwin lbw for 25, with the 20-year-old all-rounder, who made his England debut in June, now having been on the winning side in all four of his Test appearances.
This result, however, meant India had won just one of their last nine Test series outside Asia -- and that against a struggling West Indies.
- 'Relentless -
But barring an innings and 159-run defeat in the second Test at Lord's, they have not been outclassed and Kohli said: "We definitely won't throw in the towel in the last game."
After they were dismissed for 271 in their second innings, England made early inroads into India's top order.
KL Rahul was bowled for a duck by a Stuart Broad delivery that kept cruelly low.
James Anderson, wicketless in an India first innings of 273 that featured Cheteshwar Pujara's excellent 132 not out, then struck twice.
He had Pujara lbw for just five on Sunday and his next over saw Shikhar Dhawan (17) well caught in the gully by Ben Stokes to leave India in the dire position of 22 for three.
"We didn't get the start that we wanted," said Kohli. "I would have to say England were relentless with the ball."
Anderson's double strike left the paceman on 559 Test wickets, just four behind retired Australia great Glenn McGrath's mark of 563, the most taken in Tests by any fast bowler.
India were again looking for another major innings from Kohli, who had already scored two hundreds this series and would pass 500 runs for the campaign during Sunday's innings.
England thought they had the world's number-one ranked batsman lbw to Ali when Kohli was on nine and 15.
But both original not out calls were upheld on review.
Kohli went to fifty with just a third four in 114 balls when he clipped Anderson off his pads.
It seemed as if England had missed a huge chance when Alastair Cook dropped Kohli off Ali at short leg.
But the very next ball saw Kohli get a thin glove and this time Cook made no mistake.
The scale of India's task was evident from the fact they had only three times before scored 200 or more in the fourth innings to win a Test match outside of Asia.
And for all that England had suffered 203-run thrashing by India at Trent Bridge last week, it was still a decade since they had last lost successive home Tests following back-to-back defeats by South Africa in 2008.
Three balls after tea, Hardik Pandya was caught for a duck in the slips by Root off Stokes.
His exit sparked a cascade of wickets, with Rahane's painstaking 159-ball effort ending when he was lbw to a sharply turning Ali delivery.
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First Published: September 3, 2018, 9:05 AM IST