Williams ripped the ball off Courtney Lawes on halfway before re-gathering his own kick ahead for a sensational individual try.
Leigh Halfpenny's conversion put Wales 19-12 ahead but there was more drama to come as England came within an inch or so of rescuing a draw right at the death.
David Strettle crossed in the far right-hand corner with the clock already having ticked past 80 minutes but the television match official ruled he had been unable to ground the ball under pressure from Halfpenny, Jonathan Davies and George North. The decision took what seemed to be an age to arrive, with the eventual verdict centreing on the footage being inconclusive.
England had led 9-6 at half time thanks to three penalties from the impressive Owen Farrell who made his first start in the No10 shirt having won his two previous caps in the centre.
A fourth penalty from Farrell had pushed the home side six points clear on 44 minutes with Wales suffering an even bigger blow in the process. Having seen his clearance kick charged down by Mouritz Botha, Rhys Priestland deliberately killed the ball and was duly yellow carded by referee Steve Walsh.
But, unlike in 2010 when Alun Wyn Jones' sin binning cost Wales dear at the same ground, Warren Gatland's men survived the 10-minute period with 14-men without conceding another score.
A third Halfpenny penalty even saw them reduce the gap to three on 51 minutes and the 2009 Lion went on to level the scores with just eight minutes remaining.
Williams had blown a glaring overlap just before Halfpenny's final three-pointer but the young Scarlet made amends with the crucial contribution on 74 minutes. Halfpenny's straightforward conversion left England requiring a converted try to tie things up and they almost did so when Toby Flood's looping pass found Mike Brown and the replacement full back shipped the ball on to Strettle. But it wasn't to be for the reigning champions who suffered defeat for the first time under interim head coach Stuart Lancaster.
Earlier on in the day, Ireland thrashed Italy 42-10 at the Aviva Stadium to secure their first win of the campaign.
Paul O'Connell and co scored five tries in total, with Lions Test star Tommy Bowe grabbing a brace and fellow Lion Keith Earls also crossing.
Prop Tom Court powered over from close range before Andrew Trimble completed the scoring with a fine 50-metre break with the final play of the game.
Italy fought hard for the opening 40 minutes and were level shortly before the break when Sergio Parisse matched an early Earls effort.
But Bowe's first try handed the home side a 17-10 lead at half time and, after Italian fly-half Tobias Botes missed a kickable penalty shortly after the restart, it was all Ireland for the remainder of the match.