The Hornets had led 2-0 at Vicarage Road with Javier Acuna and Davide Faraoni both scoring well-taken individual strikes either side of half-time to put the Sky Bet Championship outfit on the verge a minor upset.
But the visitors hit back with 15 minutes remaining as Josh Murphy continued the trend of scoring fantastic efforts with Hooper then equalising for Chris Hughton's side with the last touch of normal time.
The summer signing from Celtic then scored the winning goal of the game as he coolly finished when through on goal with just five minutes of extra time remaining.
Nathan Redmond, who has impressed in his first few outings for Norwich, forced a smart save out of Jonathan Bond in what proved to be the only real effort of a languid opening 20 minutes.
With both managers making eight changes to their respective sides it took the game a while to burst in to life but it was Paraguayan forward Acuna who struck midway through the first half to put Watford ahead.
He picked up possession on the edge of the Norwich box before firing an unstoppable shot past Norwich goalkeeper Mark Bunn, who could only get a despairing glove to Acuna's bullet.
Wes Hoolahan, one of those recalled in to the Norwich side, saw a fantastic dipping effort tipped over by Bond before Bradley Johnson cracked a header off the crossbar from the resulting corner as the visitors chased an equaliser before the break.
However, the more regular forays forward left Norwich susceptible to the counter-attack and Acuna almost benefited from a wayward header from Norwich skipper Russell Martin, but the former Real Madrid forward could only shoot wide from just outside the box.
Hughton made an attacking change at half-time by replacing full-back Javier Garrido with Johan Elmander, who scored a brace in Norwich's 6-3 second-round success at home to Bury.
But it was Watford who created a number of chances at the start of the second-half and it did not take them long to double their advantage.
Firstly, Faraoni pounced on a loose ball from Johnson before forcing a simple save out of Mark Bunn, who had to be more alert just seconds later to push Sean Murray's effort away from goal.
The former Blackburn stopper was keeping the visitors in the tie and soon had to pull off a double-save to deny Diego Fabbrini and substitute Troy Deeney before being left helpless as Faraoni's 55th-minute piledriver ripped past him to double Watford's lead.
Murray was the next to try his luck from range but was just off target with his curling effort, with Hughton introducing young winger Murphy and midfielder Leroy Fer as he strived to get his side back in to the game.
And it was academy graduate Murphy, making his first-team bow, who continued the pattern of marvellous goals and gave Norwich hope with a thunderous drive that clipped the crossbar and left Bond with no chance.
Bond did manage to claw away Fer's looping attempt three minutes later as the Watford defence started to creak under the pressure being applied by the visitors.
Martin was the next to come close to an equaliser as Bond was again called in to action to palm away his late header but Hooper went one better as he glanced Redmond's cross in to the far corner to level for Norwich with the final touch of normal time.
Deeney had the first decent chance of extra-time but his low shot was well held by Bunn with the hosts looking to respond to the body-blow of conceding such a late equaliser as Norwich won a string of corners without creating an end product as the first half of extra-time failed to separate the two sides.
Martin Olsson rode a number of Watford challenges before shooting wide and Deeney again tested Bunn with the outcome of the game still on a knife-edge.
But it was the visitors who took the lead with five minutes of extra-time remaining as Hooper timed his run to avoid the offside trap before collecting a through ball from Fer and sliding it through the legs of Bond to put the Canaries ahead for the first time.
Bunn then provided more goalkeeping heroics to keep out first Deeney and then Hornets' skipper Marco Cassetti before being shown a yellow card following a goalmouth altercation as Norwich held on to advance.