Islam Feruz and Alex Kiwomya both had chances for the Blues but twins Jacob and Josh Murphy persistently threatened on the break and the latter won a spot-kick in the final minute
Cameron McGeehan's last-minute penalty gave Norwich the edge over Chelsea in the FA Youth Cup final after an entertaining first leg at Carrow Road.
The Blues, who are the current cup holders and also took the crown in 2010, are regarded as favourites to retain the trophy but were frustrated by the determined and organised home side.
Islam Feruz and Alex Kiwomya both had opportunities to break the deadlock but Chelsea struggled to turn their dominance of the ball into clear-cut chances and were beaten at the death when McGeehan, playing against his former club, slammed home his spot-kick after Alex Davey had fouled Josh Murphy.
The Canaries enjoyed the encouragement of 21,595 supporters on home turf but the threat posed by the pacy Chelsea front three quickly became apparent in the early stages.
Jeremie Boga was denied a shot on goal by a superb last-ditch challenge by Kyle McFadden before Kiwomya raced into a similar position in the inside-left channel only to be flagged for offside.
The third member of the Blues' forward line, Feruz, was presented with arguably the best chance of the match when Kiwomya cut a low cross back from the right flank but the Somalia-born Scotland youth international blazed over the crossbar.
Norwich goalkeeper William Britt was worked for the first time just inside the half-hour mark when a free-kick lofted into the box fell to Nathan Ake. The Blues centre-back was afforded two bites of the apple but after Britt had saved his first shot, Cameron Norman blocked the second effort on the line.
While Chelsea had dominated possession, however, the Canaries were growing in confidence on the counterattack thanks to the efforts of the promising Murphy brothers. After a short corner, Jacob's cross just evaded two yellow shirts at the back post and Josh was similarly close to picking out Carlton Morris after a darting run.
Despite those scares, the Blues ended the half in the ascendancy and troubled Norwich for the second time from a set piece. Charlie Colkett's corner from the right was headed goalwards by Feruz from point-blank range but Cameron King cleared off the line and the hosts successfully marshalled the loose ball to safety.
The second period soon fell into a similar pattern of Chelsea control but as the visitors pushed further and further into the Norwich half, with Ake often stepping out of defence to join the attack, gaps began to appear at the back.
Josh Murphy stepped inside from the wing and drew right-back Ola Aina out of position before releasing Morris on the left, and the striker shot narrowly over the bar after he had cut inside to shake off Davey.
Norwich retained their composure in midfield when they did get their foot on the ball and minutes later, Jacob Murphy pulled a shot into the side netting after neat work involving the winger, McGeehan and King.
Chelsea introduced attacking midfielders John Swift and Charly Musonda Jr but the Canaries centre-back pairing of McFadden and Harry Toffolo was in impressive form and their dogged display was rewarded as the final whistle approached.
Josh Murphy sped into the box to the outside of Davey and when the centre-back stumbled, he brought the winger down with him. Norwich captain McGeehan hit the roof of the net to spark jubilant celebrations.
Norwich: Britt; Norman, McFadden, Toffolo, Wyatt; McGeehan, Randall; Jacob Murphy, King (Young 74), Josh Murphy; Morris.
Subs not used: Reading, Browne, Hodd, Lokko.
Chelsea: Beeney; Aina, Davey, Ake, Wright; Baker, Loftus-Cheek; Kiwomya, Colkett (Swift 60), Boga; Feruz (Musonda 73).
Subs not used: Killip, Dabo, Conroy.