India, resuming on their overnight 112 for four, were bowled out in under 25 overs on Thursday for 178 before lunch on the final day in pursuit of what would have been a new fourth innings record winning total of 445.
Victory saw England end a winless Test sequence stretching back nearly a year to their defeat of Australia at Durham in August 2013.
Off-spinner Moeen Ali did the bulk of the damage with Test-best figures of six for 67, including a spell of four for 17 in 22 balls on Thursday, the all-rounder ending the match by bowling India last man Pankaj Singh.
The series continues with the fourth Test at Old Trafford starting on August 7.
Brief scores
England 1st Innings 569-7 dec (I Bell 167, G Ballance 156, A Cook 95, J Buttler 85; B Kumar 3-101)
India 1st Innings 330 (A Rahane 54, MS Dhoni 50; J Anderson 5-53, S Broad 3-66)
England 2nd Innings 205-4 dec (A Cook 70 no, J Root 56; R Jadeja 3-52)
India 2nd Innings 178 (A Rahane 52 no; M Ali 6-67)
Result: England won by 266 runs
Series: Five-match series level at 1-1
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