The off-spinning all-rounder took four wickets on the final day to finish with impressive second-innings figures of 6-67, as England completed a comprehensive triumph before lunch on the final day at the Ageas Bowl.
In the process, the hosts ended a run of 10 Test matches without a win in the five-day format, their last success having come against Australia in Durham last August.
The victory is sure to alleviate some of the pressure that had mounted on captain Alastair Cook after his side had fallen 1-0 down in the series in the second Test at Lord's.
His knocks of 95 and 70 not out helped England set India a target of 445, which would have represented a world-record chase.
The tourists began Thursday's morning session on 112-4 and were further hindered when man of the match James Anderson (2-24) struck two early blows.
Anderson first tempted Rohit Sharma (6) to edge behind, before India captain MS Dhoni (6) fell in similar circumstances to leave India reeling on 120-6.
Moeen then came to the fore, taking three wickets in the space of two runs to put England on the brink of victory.
Ravindra Jadeja (15) and Mohammed Shami (nought) were bowled either side of Bhuvneshwar Kumar (nought) being held at gully by Anderson after inside-edging onto his pad.
Moeen soon finished the job, bowling Pankaj Singh (nine) to claim a fourth wicket in the space of 22 deliveries.
England's players celebrated the end of their drought in a tight huddle, and will now look to take the lead in the series at Old Trafford when the fourth Test starts on August 7.