Wes Foderingham and Antonee Robinson both turned the ball into their own net, but Bobby Decordova-Reid and Willian registered at the correct end of the field in an all-action second half, leaving Paul Heckingbottom's ravaged side still searching for their first three-point haul of the new season
Owing to an improved second-half display against Chelsea on Monday, Marco Silva handed starts to Carlos Vinicius, Decordova-Reid and Alex Iwobi, while Auston Trusty and Basham were brought into a depleted Blades side alongside Oliver Norwood.
The Sheffield United backline came under immediate pressure from Fulham, and ex-Arsenal man Trusty had to deflect Decordova-Reid's goal-bound strike over the top with just six minutes gone.
Foderingham continued to receive adequate protection, but the Blades number one was called into action in the 19th minute, beating away Willian's attempt after Anel Ahmedhodzic lost the ball to Vinicius in a dangerous area.
Four minutes later, Sheffield United defender Basham - who had come in for the injured John Egan - attempted to send a cross into the box, but the 35-year-old's left ankle buckled and play was halted for 12 minutes as medics attended to the stricken centre-back.
Fulham captain Tim Ream comforted his opposite number while several players turned away from the nauseating sight, and Basham was given oxygen before being stretchered into an ambulance to a standing ovation from both sets of supporters.
Following the restart, Fulham picked up where they left off and continued to drive at the Sheffield United backline, which had been reinforced by the introduction of Jack Robinson for Basham as 14 minutes of added time went up on the fourth official's board.
Andreas Pereira attempted to pick out the top corner from range in the eighth minute of the 14, but the Brazilian's effort flashed just wide as the Blades went in for a much-needed breather with their clean sheet intact.
An open start to the second period saw Oli McBurnie head straight into the arms of Bernd Leno in the 48th minute, but Fulham finally wore down the Blades and drew first blood six minutes later.
The Cottagers had experienced plenty of joy down the left flank already, and a quick break down that side ended with Pereira fizzing a gorgeous ball into the box for Decordova-Reid, who kept himself onside and clinically found the bottom corner.
However, Sheffield United levelled in unorthodox circumstances just 14 minutes later, as an injured Issa Diop gifted possession straight to a yellow shirt inside his own half, but rather than put the ball out of play, the visitors surged forward and levelled as Antonee Robinson turned Yasser Larouci's low cross into his own net.
Diop was immediately replaced by Tom Cairney for an indignant Fulham, but Craven Cottage felt that justice was served in the 76th minute, and the captain had only been on the field for five minutes before avenging his injured teammate.
Cairney slipped as he let fly from range, but his looping effort struck the crossbar and bounced off the head of Foderingham into the back of the net - akin to Emiliano Martinez's freak own goal against Arsenal last season.
Fulham came agonisingly close to a third in the 83rd minute through Harry Wilson, but the Welshman's curling effort struck the frame of the goal, and Sheffield United were given eight minutes of added time to salvage something from the game.
Instead, the Cottagers put the tie to bed in the second additional minute, as Willian found a yard of space inside the area and unleashed a low drive that proved too hot for Foderingham to keep out.
Prior to Nottingham Forest's meeting with Crystal Palace, Fulham lie 12th in the rankings ahead of their trip to Tottenham Hotspur on October 23, while 20th-placed Sheffield United seek their first win at the ninth attempt when they host Manchester United following the international break.