Young Canaries fight back from 4-1 down
The Canaries trailed 3-1 at the break as two goals from Mitchell Curry and a further Harrison Chapman strike emphatically cancelled out Todd Cantwell's 7th minute penalty.
Boro extended their advantage with just over an hour gone when Lee Hetherington flicked home from a corner, before Benny Ashley-Seal reduced arrears with a fired finish.
Glenn Middleton's composed effort put City within touching distance, ahead of Louis Ramsay's majestic volley with just three minutes left of normal time sparking jubilant celebrations and ensuring extra-time was required.
With chances at a premium during the additional 30 minutes, it would be Cantwell who provided the decisive moment by poking into the net from deep inside the area on 103 minutes to complete the most remarkable of fightbacks and send his side through.
City had begun the game brightly and were given the chance to go in front early on when Cantwell's trickery saw him sent to ground in the box by Niall McGoldrick, consequently earning his side a penalty.
Cantwell himself stepped up to take the spot-kick, confidently rolling the ball low into the corner to put the hosts in front after just seven minutes.
The Canaries' lead was to be short-lived however; a low delivery from the Boro right picking out Curry who swiveled before sending low beyond goalkeeper Aston Oxborough with ten minutes played.
City looked to respond, and smart work on the edge of the box from Jamal Lewis arguably deserved more as he created space for himself well ahead of firing over.
Instead it would be the visitors who struck again midway through the half, with Curry again providing the telling touch by superbly bringing down Arnel Jakupovic's direct pass and lashing home.
Boro quickly extended their advantage, with Chapman finding room on the right-hand side of the box to angle a low drive beyond Oxborough and into the corner.
Graeme Murty's youngsters did well to weather the storm, gradually drawing themselves back into the contest but being unable to make inroads into the two-goal deficit before the break.
Such efforts continued after the restart, but Boro were largely successful in taking the sting out of the game as it approached the hour-mark.
Having contained City, they then looked to cause problems of their own, with substitute Patrick Reading seeing a rasping drive cannon off the crossbar after Oxborough's fine low save denied Chapman.
From the corner which followed Boro pressed matters home; Hetherington sweeping Chapman's set-piece into the net on the volley to make it 4-1.
Despite that setback, the Canaries continued to threaten, and deservedly scored their second when Ashley-Seal expertly met a cut-back from the left on 69 minutes.
Buoyed by that goal, Murty's side began to press with greater authority; Ashley-Seal drawing a save from Aynsley Pears before Cantwell blasted wide on the run.
Not to be denied, the lively Middleton got the goal his impressive personal display deserved with ten minutes remaining by striking low into the corner to put City firmly back in contention.
Middleton was again involved moments later, bursting well down the left flank to test Pears having been picked out well by Cantwell following a neat turn in midfield.
City were rampant, and got a fully merited leveller on 87 minutes with one of the finest goals likely to be seen at Carrow Road this season.
A corner was cleared only as far as Ramsay outside the box; the defender quickly sending back with interest on the volley to crash home off the crossbar in stunning fashion and send the game into extra-time.
Having earned extra-time, the Canaries then went in front on 103 minutes when Cantwell prodded home from close-range - some 96 minutes after scoring the opening goal in what was proved to be a remarkable game.
That goal was enough for Murty's battling youngsters to book their places in the quarter-finals, setting up a last-eight tie with Manchester City at The Etihad Stadium in the process.
FULL-TIME: Norwich City 5-4 Middlesbrough (AET)
GOALS: Canaries: Cantwell (pen. 7, 103), Ashley-Seal (69), Middleton (80), Ramsay (87). Boro: Curry (10, 22), Chapman (25), Hetherington (61).
NORWICH CITY: Oxborough; Ramsay, Crowe (c) (Sambu, 96), Odusina, Syme; Pollock (McIntosh, 60); Cantwell, Godfrey; Lewis, Ashley-Seal (Couzens, 80), Middleton. Subs not used: Hale-Brown (GK), Fleming.