Hooper sent the tie into extra-time with an injury-time equaliser before then producing a moment of class to beat Watford keeper Jonathan Bond in the 115th minute to seal the Premier League side's passage to round four.
Goals from Javier Acuna and Davide Faraoni were almost enough to give Watford the win, but Norwich fought back to level through Josh Murphy and Hooper, before the former Celtic striker had the final say as penalties loomed.
Watford found their rhythm after the goal, but McEachran’s debut was then cut short due to a back injury and, moments later, Bond had to be sharp to tip Wes Hoolahan's shot over before Bradley Johnson saw his header cannon off the bar from the resulting corner.
Hughton introduced Johan Elmander for Garrido at the break, but Watford started the half on the front foot and Zola's side got their just reward when Faraoni doubled their advantage with a sublime effort.
Debutant Murphy then showed his confidence by pulling one back for Hughton's men 13 minutes from time as he drilled in off the post from the edge of the area.
And Norwich's pressure then paid off as Hooper headed home Redmond's cross in the 93rd minute to force the additional 30 minutes.
