Spinner Moeen took six for 67, the first five-wicket haul of his test career, and India crumbled to 178 all out in their second innings on the final morning of the match.
England took the six wickets they needed to secure a first test win since they beat Australia at Durham last August, ending a dismal run of 10 matches without victory.
"It was a bit of a scramble for the souvenir stumps at the end as it was some people's first win and I want them to remember that," England captain Alastair Cook said at the presentation ceremony.
"But the senior players stood up as well, the team was fantastic from one to 11.
"Moeen Ali's bowling has come on leaps and bounds this summer, he's fronted up to the responsibility, worked really hard and responded well. Getting six wickets to win the game, you couldn't ask for more."
Moeen dismissed Ravindra Jadeja, Bhuvneshwar Kumar and Mohammed Shami in the space of 12 balls, before he aptly took the final scalp of Pankaj Singh to complete figures of six for 67.
Earlier in the morning, James Anderson, who on Friday faces a hearing into his part in an altercation with Jadeja at Trent Bridge, had got England off to the perfect start by dismissing Rohit Sharma and M S Dhoni.
India batsman Ajinkya Rahane played with composure to reach his second half-century of the match, but his unbeaten 52 was in vain as England maintained the dominance they had shown over the first four days.
Man-of-the-match Anderson drew Sharma into a faint edge through to wicketkeeper Jos Buttler before dismissing Dhoni in similar fashion with a ball that nipped away from the right-hander.
Cook made a crucial bowling change, by introducing Moeen who bowled Jadeja with a full-length delivery before inducing Kumar, who has three half-centuries in the series, to edge on to his pad and the ball looped up to Anderson at gully.
Moeen continued to bamboozle the Indian batsmen and he bowled Shami and Singh to wrap up a deserved victory for the hosts.
"We'll have to think about if for the next Test as the extra batsman didn't really score anything - a lot will depend on the kind of wicket that's provided but I don't expect it to be different to the wicket here," India captain Dhoni said.
"(Fast bowler) Ishant Sharma definitely won't be fit for the fourth test. Pankaj Singh bowled really well, he was the one person bowling the right length and he was unlucky not to get wickets. Whenever I asked him to bowl he gave it everything."
The fourth test begins at Old Trafford, Manchester on Aug. 7.