Both Sean Dyche and Scott Parker decided to make a lot of changes to their line-ups and it was the Burnley back-ups that stood up to the challenge.
A double from Jay Rodriguez and a late goal from Kevin Long was enough to get the job done for the East Lancashire outfit. It's only the third time that Burnley have scored more than once this season, and the first time they have scored more than two.
Vydra was very lively in the opening stages and he forced Marek Rodak into a smart save after turning inside Michael Hector with ease.
Burnley were doing all of the probing and Rodriguez saw his shot from the edge of the box blocked, after being neatly teed up by Jack Cork.
Phil Bardsley's optimistic strike from distance was spilled by Rodak but Vydra snatched at the opportunity, dragging his shot across the face of goal.
Fulham were struggling but they thought they had taken the lead, against the run of play, through Aleksandr Mitrovic.
The Serbian striker nodded into an empty net after latching onto Aboubakar Kamara's flick. However, Kamara had strayed offside when he made contact with the ball. A lucky escape for the Clarets.
The momentum didn't turn, though, and Vydra fired tamely into the body of Rodak after being played in on goal by McNeil.
Eventually, Burnley's efforts were rewarded through Vydra's striking partner.
Cork whipped a beautiful delivery into the box and Rodriguez was positioned well, stooping low to head the ball into the far corner from close range.
It was no more than Burnley deserved and Johann Berg Gudmundsson also tried to get in on the act, fizzing a powerful strike over the bar after James Tarkowski's cross was only half-cleared.
Fulham came out of the blocks firing and Bailey Peacock-Farrell was called into action.
The Irishman sprung to his right to keep out Joe Bryan's powerful half-volley, before getting up quickly to deny Mitrovic. Mitrovic retained the ball and worked himself another angle but Peacock-Farrell was down smartly to keep hold of the low shot.
Fulham were better in the second period but they struggled to conjure up many decent openings.
In fact, it was Burnley who created the next big opening. Dale Stephens saw his fierce strike blocked by Tosin Adarabioyo after Vydra pounced on a mistake from Tim Ream and rolled the ball back.
It wasn't long until Vydra's next direct run, this time drawing a clumsy foul from Hector as he weaved into the box. A sloppy penalty kick to give away.
Up stepped Rodriguez, who struck the ball down the middle for his second goal of the game. Game over? Not quite.
Substitute Ruben Loftus-Cheek threaded Kamara in beyond the Burnley defence but Peacock-Farrell got a vital glove on the ball. It slowed down the momentum of the shot and Tarkowski was able to get back and steer the ball to safety. A big chance that went begging for the hosts.
It just wasn't going to be Fulham's day and their shaky defence was breached once again as they pushed more men forward.
Gudmundsson's hooked cross was nodded across for Rodriguez, who turned the ball towards Long, and the big central defender confidently dispatched the ball from point-blank range.
Burnley will take on either Bournemouth or Crawley Town in the fifth round of the FA Cup.
Fulham: Rodak 6; Hector 6, Ream 5, Adarabioyo 6; Reid 7, Lemina 7, Onomah 5, Anguissa 6, Bryan 7; Mitrovic 6, Kamara 6.
Substitutes: Loftus-Cheek 6, Kebano 6, Odoi n/a, Carvalho n/a.
Burnley: Peacock-Farrell 7; Bardsley 6, Long 7, Tarkowski 7, Pieters 6; Gudmundsson 7, Cork 8, Stephens 7, McNeil 7; Vydra 7, Rodriguez 8.
Substitutes: Benson n/a, Dunne n/a, Lowton n/a, Mumbongo n/a.
Bobby Decordovo-Reid (Fulham): Gave Pieters a challenge on the flank but he was withdrawn on the hour mark. Parker was probably looking to protect one of his key players for the league.
Jay Rodriguez (Burnley): Wasn't as involved in the match as Vydra but his contributions were more obvious. A poacher's opener, a clinical penalty and the illustration of good awareness for Burnley's third.