After Leigh Halfpenny’s boot gave Wales a 9-3 lead at half time, Cuthbert’s second-half try gave Wales daylight after a turnover in the middle of the park.
Cuthbert put the game beyond England’s reach after fantastic work from man of the match Justin Tipuric, who drove forward before an easy offload for the winger.
Wales’ victory gave them their fourth title in the Six Nations era, and their third in the last six years.
For Stuart Lancaster’s England, a strong campaign came to a clattering end, with many of his star performers throughout the tournament struggling to deal with the intensity of the Welsh.
And Wales showed their strength from the first whistle, with two Halfpenny penalties giving them a 6-0 lead, with England struggling to cope in the scrum.
The first half failed to present too many open chances for a try, but what was lacking in this department was made up for intensity.
George North nearly broke through after Dan Biggar’s interception, but a fantastic try-saving tap tackle from Dan Cole saved England’s blushes.
The closest England came to a try in the first 45 minutes came through Mike Brown, who escaped the Welsh defence but saw his inside pass plucked out of the air by Toby Faletau.
Owen Farrell halved the lead with a rare penalty and the visitors looked to be back in the game, but the brilliant Halfpenny again made England pay with a 45-yard penalty from a tight angle.
The second half started slowly, but Wales still emerged on top in the aerial battle and in the scrum, and after five minutes of sustained pressure went 12-3 up through Halfpenny’s penalty.
But Wales made the decisive breakthrough via Cuthbert, who went over in the corner after replacement Ken Owens turned the ball over from Tom Wood in midfield.
Owens, who had been on the pitch for just over a minute, broke quickly, and three passes later saw Cuthbert race over despite Mike Brown’s attempts to stop him.
Wales then added to their lead, with Warburton’s break ending up with Justin Tipuric, who drove forward with the ball before offloading to Cuthbert to go over with ease in the same corner.
Dan Biggar’s boot added to Wales’ lead, and despite the numerous potential outcomes possible ahead of the encounter, gave the hosts a 27-point lead that looked too much for England to overturn.
In winning, the Welsh prevented England’s first Grand Slam victory in a decade, while Wales claimed their fourth Six Nations title (their 25th title overall) and their second consecutive tournament win.