Cuthbert crossed twice in a 10-minute spell midway through the second period to ensure Wales turned a 9-3 half-time lead into a tournament-winning success as England failed to notch a clean sweep at the final hurdle for the second time in three years.
Ospreys openside Justin Tipuric did his Lions chances the world of good with a man-of-the-match display as Wales won the battle of the breakdown.
Tipuric set up both Cuthbert's scores, with his well-timed pass allowing the right wing to fend off Mike Brown for the first, before a beautiful show and go and a telling turn of pace created the second after fellow flanker Sam Warburton had initially burst through in midfield.
The ever-reliable Leigh Halfpenny added four penalties, with Dan Biggar slotting a conversion, drop goal and penalty of his own after Halfpenny took a second-half knock.
Wales began brightly in front of a 74,000 sell-out crowd at the Millennium Stadium as Halfpenny kicked them in to a 6-0 lead with 11th and 17th-minute penalties.
Owen Farrell struck back as the game hit the quarter way mark but it was to be the only time England troubled the scoreboard in a disappointing showing from a team that had already seen off Scotland, Ireland, France and Italy since early February.
England failed to fire as the majority of the crowd sensed it would be a Welsh day to remember just six weeks after a horror first 43 minutes against Ireland brought their own Grand Slam aspirations to a premature end.
The first period didn't feature any tries but it was hugely entertaining nonetheless as both sides kept the ball in hand at almost every opportunity.
Manu Tuilagi's dropped pass 30 metres out saw England's biggest chance go begging but the visitors had Brown to thank for a try-saving tap tackle that somehow stopped George North from streaking away on halfway.
The second period started at a slower pace but Wales deservedly moved to the top of the table as things stood when Halfpenny stretched the gap to nine points at 12-3 11 minutes after the break.
The pivotal moment then arrived five minutes later as England-born Cuthbert cut clear down the right to send the stadium wild.
Halfpenny was off target with the attempted extras but Wales were celebrating again nine minutes later as Biggar kept his composure to ensure Welsh pressure paid off when he sent over a snap drop goal.
Fast forward just 90 seconds or so and Wales had effectively put the game, and the Championship, to bed thanks to a second Tipuric-Cuthbert combination down the right.
This time it was Warburton who put England on the back foot by powering through in midfield. The man who led Wales to Grand Slam success last season was brought down 40 metres out but Wales recycled quickly and Tipuric showed his class and speed by cutting through, dummying Brown and handing Cuthbert the simplest of touchdowns.
England refused to give up the ghost as play continued to move from end to end but Wales were on a high and never showed any signs of taking their foot off the gas in the final 15 minutes. Biggar's well-struck conversion made it 24-3 after 66 minutes before a 71st-minute penalty extended the lead to 27, pushing them past the 25-point margin of their previous biggest win over England way back in 1905.