Two goals from Chelsea's Jeremie Boga were not enough to prevent City's youngsters making it a double celebration following the senior team's successful fight for Premier League survival.
Boga wiped out the visitors' first-leg lead early on, but Adam Nditii deflected a Joshua Murphy cross into his own net to put Norwich back in front.
Islam Feruz missed a great chance for the hosts before Cameron McGeehan - once on Chelsea's books - scored from the penalty spot as he had in the first leg to give City breathing space.
And the game was up for the home side when Joshua Murphy capitalised on good work from his twin brother Jacob to make it 3-1 on the night.
Boga reduced the deficit, playing a one-two with Charly Musonda and finishing well, but with three minutes remaining it was too tall an order for Chelsea to avoid defeat.