Matt Doherty and Raul Jimenez scored in either half of normal time to put Wolves in command of the all-Premier League affair, but Gerard Deulofeu came off the bench to turn the tie on its head, scoring a world-class goal with 11 minutes to go to set up a dramatic finale.
Watford's pressure was relentless and it told as Troy Deeney was tripped by Leander Dendoncker in the final seconds of the match at Wembley, allowing the Hornets' skipper to brush himself down and convert for extra time.
Deulofeu then scored the decisive fifth goal of the match in the additional 30 minutes of time, becoming the seventh player to score a brace in the semi-finals of the FA Cup and setting up a showdown with Manchester City here in a month's time.
There was very little between the sides in the opening half-hour of the match, with both teams managing a couple of off-target attempts apiece.
Watford then carved out a good opportunity for Andre Gray, preferred to Deulofeu alongside Deeney from the off, as he got on the end of his strike partner's deep delivery but hooked the ball over the crossbar with just John Ruddy to beat away.
Just when the Hornets were starting to build some momentum, though, with their diamond formation in midfield causing problems for Wolves, their opponents opened the scoring through Doherty.
The wing-back got away from Abdoulaye Doucoure at the back post and nodded home Diogo Jota's curled cross into the six-yard box after Wolves' corner - coming after Dendoncker's deflected shot was pushed over the bar - was worked short.
Wolves nearly scored a second in a matter of minutes as Jota, who grew into the first half, grazed the outside of the post with his curled effort.
Javi Gracia's men should have gone into the interval on level terms, however, as Gray missed his second good opportunity of the half by failing to convert Deeney's knockdown, though credit has to go to Conor Coady for diving in front of the goal-bound shot.
Doucoure flicked the ball over and Jimenez nearly nearly squeezed a shot in in from a tight angle as the two sides continued to trade blows in the opening 15 minutes of the second half.
After plenty of toiling from the Hornets, Wolves scored what looked to be a crucial second goal thanks to a brilliant finish from Jimenez, chesting Doherty's long ball and acrobatically sending it past Heurelho Gomes - VAR confirming he was just about onside.
Watford were given a route back into the game 10 minutes from time through Deulofeu's brilliance, with the substitute curling the ball over Ruddy from a standing start to net from his side's first shot on target, Ruddy only able to stand and watch as the ball found the top corner of his net.
Gracia's side dominated possession in the remainder of the match, though, and they were rewarded for their pressure as Dendoncker stepped across Deeney, got none of the ball and the striker beat Ruddy with the power on his penalty.
Having picked themselves up from the double setback, Wolves looked to be in control of the game in extra time as Jonny had a well-struck shot tipped over.
Through a rare attack, though, Gray slipped the ball through for Deulofeu and the super sub coolly slotted it back across Ruddy into the far corner.
Wolves were unable to muster another shot in the remainder of extra time, with Ivan Cavaleiro rounding Gomes but failing to keep the ball under control from the best of their openings, meaning just a second ever FA Cup final appearance for Watford to look forward to after the most dramatic of ties.
WATFORD (4-4-2): Gomes; Femenia (Janmaat 108'), Mariappa, Cathcart, Holebas (Masina 98'); Hughes (Deulofeu 66'; Sema, 112'), Doucoure, Capoue, Pereyra; Deeney, Gray
WOLVES (3-5-2): Ruddy; Saiss, Coady, Boly; Doherty, Dendoncker, Moutinho (Traore 102'), Neves (Bennett 88'), Jonny (Vinagre 105'); Jimenez, Jota (Cavaleiro 89')
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