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Italy had held a shock 10-5 lead at half-time.
England prop Dan Cole was driven over for a try off a close-range line-out to open the scoring.
Italy fly-half Tommaso Allan, who had previously missed two penalties, reduced the deficit with a well-taken drop-goal and when a third penalty came back off the post, wing Giovanbattista Venditti was first to the loose ball for a try that stunned Twickenham.
Allan converted and Italy led an England side, who had won all 16 of their previous Tests, by five points.
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But the home side, who have won all 22 of their previous Tests against the Azzurri, opened the second-half with a quick brace from Danny Care and Elliot Daly.
Michele Campagnaro scored a fine individual try to maintain Italian pressure before Jack Nowell (2) and Ben Te'o crossed the Italian whitewash to blow out the final scoreline.