The game was poised on a knife-edge at 12-12 when Williams stripped Courtney Lawes of possession and chipped forward from midfield, the ball bouncing kindly into his hands for him to score under the posts.
England thought they had crossed at the death through David Strettle, but the combination of Jonathan Davies and Leigh Halfpenny did enough to deny him, although referee Steve Walsh’s decision to defer the decision to the video referee led to a nervous few moments for all inside Twickenham.
Owen Farrell and Halfpenny shared eight penalties apiece and each missed one.
But that told only half the story of a thrilling game, full of hard hits and endeavour from two sides who refused to give an inch.
Farrell was superb in the No 10 shirt for England, as was Manu Tuilagi at outside centre, while Sam Warburton was equally impressive for the Welsh.
Wales travel to Italy and host France as they look to add a grand slam to the Triple Crown they won here.
Stuart Lancaster’s England now travel to Paris.
Read Nick Westby’s Twickenham report and reaction in Sports Monday, free with Monday’s Yorkshire Post newspaper.