The Canaries were defending a slender 1-0 advantage from the first leg following a stoppage-time penalty from captain Cameron McGeehan, who was once on Chelsea's books.
The Blues wasted little time in wiping that lead out as Jeremie Boga swept them ahead on 15 minutes.
However, an own goal from Adam Nditi, diverting in Joshua Murphy's cross, put Norwich back in front on aggregate, with another spot-kick from McGeehan making it 2-1 on the night before half-time.
After Islam Feruz had missed a great chance to equalise, Norwich scored again on the break as Joshua Murphy converted a low cross from his twin brother Jacob to put the tie beyond Chelsea, despite a late goal from Boga.
The Blues had started full of purpose and it was no surprise when they took the lead on 15 minutes when Boga drove to the edge of the penalty area before slotting a low shot past Will Britt.
Norwich, though, were soon level after Joshua Murphy collected a short corner and cut the ball back across goal, where it cannoned off Nditi and beat Mitchell Beeney at his near post.
The Blues responded as first Alex Kiwomya, the tournament's leading scorer, stabbed a low effort wide, and then a 25-yard free-kick from captain Lewis Baker brought a fine save from Britt.
Feruz then put a great chance wide after being played in by Baker - and it proved costly.
Norwich striker Carlton Morris carried the ball into the penalty area, where he was nudged over by Alex Davey, and it was enough for referee Neil Swarbrick to point to the spot.
McGeehan - released by Chelsea after being bought for £80,000 from Fulham - kept his cool again to drill the ball into the net and put Norwich in command.
Adi Viveash's young Blues again pressed at the start of the second half, with a deep cross from Ola Aina falling nicely for Baker, but his acrobatic effort was turned away by the Norwich keeper.
On the hour, Feruz was played in behind the visitors defence, but Britt was out quickly to make a superb sliding block, with the Blues striker then firing the rebound wide.
Once more it proved a telling miss as Norwich wrapped up the trophy with a breakaway goal on 76 minutes.
Cameron King fed Jacob Murphy down the left and his low centre was drilled in from six yards by his twin brother Joshua.
A superb last-ditch tackle from Chelsea substitute Charly Musonda denied Jacob Murphy rolling in a fourth on the break.
Boga slotted in a second with three minutes left after a neat one-two with Musonda, but Neil Adams' men closed out for a famous victory.