The Blades ought to have had victory wrapped up by half-time, but Oli McBurnie was unable to give their flowing football the flourishes it deserved. The Cottagers would have made the most of their reprieve had Aleksandar Mitrovic been sharper in front of goal.
As it was, both sides took their first point of the season and were left wishing it had been more.
There have been plenty of hard-luck stories for Sheffield United already this season, and an early injury to Max Lowe provided another. Ultimately, though, this was a day when they failed to help themselves.
They were even awarded a penalty by Andre Marriner with the help of the VAR monitor. Billy Sharp smashed it home emphatically, showing Mitrovic how it was done after the Serbian had clipped the crossbar from 12 yards, but in open play they lacked that ruthlessness.
Until Marriner and Sharp's intervention, it had looked as if the Blades were heading for another defeat, Ademola Lookman's brilliant goal making the most of Fulham's much-improved second half, so there was something to be grateful for, but still the over-riding feeling was of frustration.
The eagerness of Scott Parker to play ballboy, running into Sheffield United's technical area and keep the flow of the game going showed Fulham had started the better, Tom Cairney stinging Aaron Ramsdale's palms from distance, but the Blades were soon into their stride and took control.
The problem was, they lacked the finishing to make the most of it, with the recalled McBurnie particularly guilty.
In an attempt to shake things up, Chris Wilder had selected Enda Stevens at left-sided centre-back, and he soon gave his side the extra thrust going forward they have been looking for in the absence of Jack O'Connell.
“Enda, higher!” John Egan had cause to shout once, but generally the latter struck the right balance.
In the tenth minute, shortly after John Lundstram had a long-range effort tipped over, McBurnie was on the end of a Stevens cross but his shot was deflected wide.
Their overlapping centre-backs are an important feature of Sheffield United's play, but it is their bluntness up front that sometimes holds them back. Chris Basham was soon putting over a delivery McBurnie glanced wide with his head. Basham also dragged a shot wide.
The plans had to be rethought inside 20 minutes, though.
Max Lowe never recovered from the hefty whack of running into fellow debutant Ruben Loftus-Cheek in the opening minutes, and when he finally accepted he had to make way, Stevens returned to wing-back with Jack Robinson coming off the bench.
An excellent passing move in tight situations released Stevens for a shot Alphonse Areloa stretched to, and he made a point-blank save when McBurnie got to another Basham cross. Sander Berge, a growing first-half influence, had a shot saved.
Robinson gave his side another line of attack, but his long throw went across the six-yard box without a touch just after the half-hour.
Although it would have been disallowed for offside anyway, McBurnie did not get a strong enough contact on another Stevens delivery.
The recalled Oliver Norwood's passing helped the quality of his side's football and it looked like he had brilliantly picked Berge out for a cross put in for an own goal, only for the Norwegian to correctly be ruled offside.
Despite throwing Rhian Brewster on for his debut after 64 minutes, the Blades never recovered that momentum after the restart, whereas Fulham were much improved.
Ramsdale made an excellent reflex save from Lookman, albeit he created the chance himself by allowing a cross through his hands.
When Robinson's hand took a corner off the head of Loftus-Cheek, Mitrovic hit his penalty against the crossbar.
After 63 minutes the Blades had another let-off, Robinson redeeming himself by getting back to block as Fulham launched a four-on-two counter-attack.
An away goal was coming, though, and Lookman provided it, bursting past Basham and Berge and smashing the ball home with 13 minutes remaining.
The Cottagers were preparing for a corner thanks to Ramsdale's fingertip save from a Mitrovic volley when Martin Atkinson whispered in Marriner's ear that he might want to take a look at an incident at the other end minutes earlier.
Upon closer inspection of the evidence, Marriner decided Mitrovic had fouled Robinson and the recently-introduced Sharp thumped the spot kick home.
Wobbling for a few minutes, Fulham finished the stronger, Norwood blocking from Lookman.
But it was Mitrovic who missed a glorious opportunity, putting the first of two headers wide with Ramsdale helpless if he had found the net two minutes into stoppage time.
It was a chance a striker of his calibre ought to have taken. It would be the story of the day.
Sheffield United: Ramsdale; Basham (Sharp 81), Egan, Stevens; Baldock, Lundstram, Berge, Norwood, Lowe; McGoldrick (Brewster 64), McBurnie.
Unused substitutes: Burke, Robinson, Verrips, Ampadu Osborn.
Fulham: Areola; Aina, Adarabioyo, Ream, Robinson; Cairney, Zambo Anguissa, Cavaleiro, Loftus-Cheek (Lemina 80), Lookman; Mitrovic.
Unused substitutes: Kebano, Rodak, De Cordova-Reid, Le Marchand, Bryan, Kamara.
Referee: A Marriner (West Midlands).
Support The Yorkshire Post and become a subscriber today.
Your subscription will help us to continue to bring quality news to the people of Yorkshire. In return, you’ll see fewer ads on site, get free access to our app and receive exclusive members-only offers.
So, please - if you can - pay for our work. Just £5 per month is the starting point. If you think that which we are trying to achieve is worth more, you can pay us what you think we are worth. By doing so, you will be investing in something that is becoming increasingly rare. Independent journalism that cares less about right and left and more about right and wrong. Journalism you can trust.