India’s second victory at the home of cricket and first since 1986 continued England’s miserable year after series defeats by Australia and Sri Lanka.
Resuming at 105 for four in pursuit of 319, Joe Root and Moeen Ali survived until the final ball before lunch when the latter fell to Sharma.
The willowy right-armer claimed four more wickets after the break as England’s batsmen capitulated in the face of India’s short-pitched barrage and were bowled out for 223, putting the visitors 1-0 up in the five-match series.
Sharma finished with career-best figures of 7-74, including five wickets in 33 balls.
Root and Moeen had earlier given hope of an unlikely home victory, having come together with England in trouble at 72 for four midway through Sunday’s evening session.
They added a further 68 runs on Monday morning before Moeen, who batted for a day to nearly save the second test against Sri Lanka at Headingley last month, fell last to a brute of a delivery from Sharma which forced him to fend off to Cheteshwar Pujara at short leg.