Tom Briscoe's try took Hull FC into a first Challenge Cup semi-final in five years in an error-strewn victory in humid conditions at Catalan Dragons.
Danny Tickle's late penalty gave the visitors a narrow half-time lead to add to tries from Hull's Shannon McDonnell and Dragons' Eloi Pelissier.
Tom Lineham, Briscoe and McDonnell all crossed after the break as the Airlie Birds extended their lead.
Morgan Escare went over for the Dragons, who could not cut the deficit.
The Black and Whites, who have won three Challenge Cups, ended a run of four straight defeats, while Catalans, who beat Hull convincingly in the Super League last month, are now left to focus on the play-offs.
Hot and sticky conditions in Perpignan seemed to impact on the handling of both teams and it took 23 minutes to break the deadlock.
Despite an aerobatic finish Damien Blanch had a try ruled out for the home side early in the game as Escare was offside earlier in the set, and Hull's better management of the ball in the first quarter paid off as they opened the scoring.
McDonnell crossed, shrugging off a string of Catalans tacklers to squeeze the ball down and video referee Steve Ganson awarded the try.
In response the home side found a rhythm, stringing together repeat sets and interchange hooker Pelissier crashed over from dummy-half to pull level after Dureau's conversion.
Both sides continue to trade handling errors, and the tight nature of the game was reflected by Dureau's successful drop goal attempt.
Tickle sent Hull in ahead on the hooter when he clipped over a two-pointer after Jamal Fakir's shot on Jacob Miller, and the second 40 remained as tightly poised as the first.
It took a few minutes to restore the Catalans lead when Escare powered onto a tidy offload in front of the posts from Lopini Paea to score.
Paea's joy turned to despair when he knocked on to gift Hull field position, and a fluid left to right move found Lineham to crash over in the corner.
Hull, who had already lost Ben Galea with a shoulder injury, saw Tickle and Aaron Heremaia clash heads but both remained on the field, and almost pulled ahead again when Danny Houghton and McDonnell ripped a hole in the Catalans line but Lineham was kept out in the corner.
However Hull were not to be denied, Houghton taking advantage of the gap left by Steve Menzies coming out of the line and his short ball to Briscoe put the England winger in for an easy run to the line.
Briscoe's score came with 15 minutes remaining but despite the Catalans onslaught, they ran in a fourth try from McDonnell to seal a place in the last four.
VIEWS FROM THE DRESSING ROOM
Catalan Dragons coach Laurent Frayssinous:
"We are very, very disappointed with the game tonight but credit to Hull FC, it was a scrappy game, but their defence was top quality. We are disappointed to have lost the game, but more disappointed that we haven't learnt from the recent months and weeks about the small details that win you the big games like today.
"When you make so many errors, when your shape is not right and when you take short cuts to try and win games it doesn't work and you get into situations like you saw us in today and you cannot win these sorts of games.
"The players are extremely disappointed in themselves tonight. There was a lot of extra effort from some of them out there and I cannot blame individuals for today, it is about learning lessons and hopefully we will learn them quickly for next week."
Hull FC coach Peter Gentle:
"I'm pretty pleased to be sat here the winner to be honest with you. Those were really trying conditions out there, they really were horrendous and the game was played at a snail's pace at times.
"Gareth Ellis, Joe Westerman and Danny Houghton all played their hearts out there for us, for Houghton to put in an 80-minute performance like that in those conditions is a pretty special effort.
"Our try line defence was superb and held up pretty well to almost everything they threw at us, although it was a slow game and that was partly down to the new balls. Once the shirts and grip got sweaty, the ball was really hard to hold in the tackle and the boys were saying it was tough to hold onto in the tackle."
Catalans: Escare; Vaccari, Duport, Millard, Blanch; Pryce, Dureau; Simon, Henderson, Casty, Menzies, Taia, Mounis.
Replacements: Pelissier, Larroyer, Fakir, Paea.
Hull: McDonnell; Lineham, Crooks, Yeaman, T. Briscoe; Heremaia, Miller; O'Meley, Houghton, Green, Ellis, Westerman, Galea.
Replacements: Lynch, Tickle, Whiting, Watts.
Referee: Richard Silverwood (RFL)