Their match at home to comfortably the best team in the Championship was 50 minutes old when the ballboy belatedly roused himself from his stool to throw the ball to Ben Osborn.
The wing-back's throw-in went to John Fleck, who fed Sander Berge, who played it to Enda Stevens, the man over, to drill home. It was all one-touch stuff, beautiful to watch, and Sheffield United were 4-0 up.
There was no rush, no panic, as the young ballboy had recognised perfectly.
Gibbs-White and Chris Basham were soon pulling out the party pieces. By the end the fans of Fulham - you know, easily the best team in the Championship over the course of the season - were singing "We've got the ball" in gallows-humour mock celebration.
The Cottagers had actually started the game looking reasonably hungry but the Blades were absolutely ravenous for the win that would book their place in the play-offs, and their makeshift 22-year-old strikers Iliman Ndiaye and Morgan Gibbs-White rose to the big occasion. Behind them, Sander Berge was in the mood too. But then everyone in Bramall Lane wearing red and white was.
The groundwork has been laid inside a rampant first half-hour.
It was clear inside the first five minutes that despite five changes, Fulham were up for what, for them at least, was a dead rubber. Bobby Decordova-Reid's shot against a defender was the clearest signal.
But so were Sheffield United.
Within 10 minutes they led, Osborn laying the ball back to Oliver Norwood, whose cross Gibbs-White headed in at the near post. In his first Championship start since November, Cottagers goalkeeper Paulo Gazzaniga seemed to misjudge the flight of the ball.
It made a ground that had started the game noisily noisier.
For just over a quarter of an hour, the Blades and their young strikers were rampant, all but putting the game to bed.
Gibbs-White won the ball from Nathaniel Chalobah and had a shot into the side-netting after good build-up play by Enda Stevens and John Fleck.
Jack Robinson headed over a Norwood corner won when Ndiaye had a shot blocked from Sander Berge's pass.
Not for the first time, Jean Michael Seri slipped and the hosts pounced, Berge finding Gibbs-White, finding Ndiaye for his fourth goal in give matches.
Gibbs-White then played like the centre-forward he has been asked to be in the absence of the injured Billy Sharp, Oli McBurnie, Rhian Brewster and David McGoldrick, flicking Robinson's header on to his fellow makeshift striker Ndiaye. Berge ran onto the pull-back and put the ball in at the near post. 3-0.
"The boys are bantering," tweeted McBurnie.
Preston North End were helping out at home to Middlesbrough but the Blades had made it irrelevant.
As half-time approached Gibbs-White had a goal disallowed for offside and Wes Foderingham stuck out a left boot to deny Fabio Carvalho what looked a certain goal. It was just another hint to a Fulham side who lost Chalobah to injury on the half-hour that there was not much point them bothering to come out for the second half.
Stevens's goal only underlined that.
Nothing after that really mattered, either at Bramall Lane or elsewhere, except perhaps where the identity of the Blades' semi-final opponents were being decided. Nottingham Forest are coming to South Yorkshire on Saturday, with the second leg on Tuesday. Huddersfield Town and Luton Town will play the other tie, on Friday and Monday.
The two strikers got rests and ovations from the crowd, as did Fleck. Not long out of Ramadan, Ndiaye again appeared to be suffering from cramp.
Gazzaniga made a good save from John Egan but was outdone as Foderingham twice leapt right late on to deny Neeskens Kebano, then Rodrigo Muniz.
Injuries or not, Forest will not fancy taking on Sheffield United's players or their fans on this kind of form.
Sheffield United: Foderingham; Basham, Egan, Robinson; Osborn, Norwood, Fleck (Hourihane 75), Stevens; Berge; Gibbs-White (Jebbison 78), Ndiaye (Osula 62).
Unused substitutes: A Davies, Bladock, B Davies, Norrington-Davies.
Fulham: Gazzaniga; Teta, Hector, Ream, Robinson; Seri (Reed 72), Chalobah (Onomah 30); Kebano, Carvalho, De Cordova-Reid; Mitrovic (Rodrigo 60).
Unused substitutes: Tosin, Cavaleiro, Fabri, Stansfield.
Referee: J Smith (Lancashire).