The Canaries' Jacob Murphy looked to have done enough to send his side through to the quarter-finals in normal time when he fired Norwich in front in the 78th minute.
However, Blues' brave youngsters refused to give up and forced the tie into an extra 30 minutes when substitute George O'Neill bagged an equaliser deep into stoppage time.
With the evenly-fought encounter heading to penalties, Norwich skipper Cameron McGeehan popped up to nod home the winner from Joshua Murphy's free-kick in the 112th minute.
It was a cruel ending for Steve Spooner’s side after a valiant effort on a chilly evening in Norfolk.
Having seen off West Bromwich Albion and Bristol Rovers in the previous rounds, Everton were waiting in the last eight for Blues if they were able to overcome the young Canaries.
And while both teams flew out of the traps, Norwich could have taken the lead inside 30 seconds only for Nick Townsend to brilliantly tip wide Cameron King’s rasping effort.
That set the tone for a pulsating encounter.
Blues pressed their opponents well and Bobby Moseley went close with a hopeful effort from distance in the 13th minute. A cleared corner fell to the left-back 35 yards out and his attempt had Norwich ‘keeper William Britt scampering back as it looped over the net.
Norwich were guilty of missing a glorious chance to take the lead in the 34th minute. Jacob Murphy’s corner from the right was spilled and fell kindly for Harry Toffolo yards out only for the defender to fire over.
Reece Hales almost made the hosts pay three minutes later. Liam Truslove kept play alive from a cleared corner with a cross from the right, Callum Thomson knocked it back and Hales hooked it goalwards only to be denied by Britt low down.
The encounter continued in the same vein after the break with Blues skipper James Fry keeping out Reece Hall-Johnson’s shot with a well-timed tackle inside the area.Both sides battled on and the Canaries got their beaks in front 12 minutes from the end of normal time with a ruthless counter-attack. The lively Jacob Murphy drove inside from the right flank and sent his shot into the far corner.
Yet it was far from game over.
Truslove almost found a way past Britt with a low drive from 25 yards before Jacob Murphy rattled the Blues bar in the 85th minute.
The pendulum continued to swing; Truslove again tried his luck with a dipping volley from the left side of the area before Townsend made a vital save to deny Norwich sub Rod Young.
With time running out, Blues’ perseverance up front paid off as O’Neill continued his run from left-back, found himself with only Britt to beat and coolly fired through the legs of the outrushing goalkeeper.
The visitors were given an early scare in extra time when Young jinked his way in from the right wing before striking the bottom of the near post.
With penalties looming, McGeehan broke Blues hearts by sending home his diving header from Joshua Murphy’s pinpoint delivery.
Norwich City U18s: William Britt, Cameron Norman, Ben Wyatt, Cameron McGeehan (C), Kyle McFadden, Harry Toffolo, Jacob Murphy, Reece Hall-Murphy, Carlton Morris, Cameron King (Rod Young 79), Joshua Murphy.
Subs not used: Kevin Lokko, Asa Swatman, Rian Carroll, Henry Randall.
Blues U18s: Nick Townsend; Dominic Bernard (Deqwon Ebanks H/T), James Fry (C), Callum Thomson, Bobby Moseley (George O'Neill 84); Charlee Adams, Liam Truslove, Reece Brown, Demarai Gray (Viv Solomon-Otabor 79); Reece Hales, Koby Arthur.
Subs not used: Callum Preston, Jean Kalenda.