The Blades are through to the last 16 of the FA Cup after a dramatic night at Craven Cottage.
In the final minute of extra-time Shaun Miller swooped to score the only goal of the game to eliminate Premiership Fulham.
The Blades more than matched the Premiership side before accounting for a second top flight club in the competition.
Clough made four changes - including the goalkeeper - for the replay at Craven Cottage. Howard, Harris, McGinn and Porter were recalled with Long, Hill and Baxter dropping to the bench, whilst Paynter was cup-tied. Also amongst the substitutes were two 18-year-old Academy prospects in De Girolamo and Dimaio.
Encouragingly, United made the brighter start and took just two minutes to force a chance - thanks to the Scottish contingent. The move began with Flynn and after Scougall had won a tackle with Parker, Murphy shot just wide from the edge of the box.
There was little for the home fans to get excited about in the opening half hour - two Tankovic shots from distance hardly troubled Howard and at the other end Maguire headed over a McGinn corner.
Dempsey fired over after Rodallega laid off Kasami's pass but the American was robbed soon after and Brayford went close to opening the scoring. Murphy, under pressure from Passley, headed wide a cross from Coady as United remained on top, although Harris was cautioned for a foul on Passley.
Close to half-time Fulham won a couple of corners but again failed to trouble Howard, the closest they came was a Burn header from Kasami's delivery but United cleared the danger.
As in the first half, the Blades began the second period with a chance - Harris curling a free-kick over from 25 yards after Amorebieta fouled Murphy. The next opportunity also dropped to United's new left-back moments later but this time he curled into the arms of Stockdale after Murphy laid off.
A spell of Fulham pressure came to nothing, prompting a double change, whilst Hangeland had to block a Scougall shot on 61 minutes as the rain began to fall in the capital. Duff's chip into the box was touched by Collins before Howard pushed around the post but the Blades were quickly back on the offensive, another Scougall shot was blocked before Brayford caused problems from the resulting corner.
Porter rolled his ankle in the first half but continued until 75 minutes when he was replaced by Baxter, who almost attached himself to a Brayford cross soon after. For the home side, shots from Tankovic and Kasami were off target.
In the final ten minutes of normal time, Baxter was off target after Scougall got the better of Dejagah and Flynn cut inside but fired over. At the other end, there was a weak penalty appeal when Dempsey fell under Maguire's challenge and Sidwell fired at Howard.
Four injury-time minutes failed to produce a clear-cut chance meaning extra-time was required, for which Hill was introduced for Harris. A Baxter corner was headed wide by Brayford as the Blades continued to match the Premiership side in additional time but cramp accounted for Scougall and Miller took over.
The best opportunity of the night arrived on 99 minutes when Brayford saw a hole open up in the Fulham defence and ran between four white shirts before drilling a low shot, which Stockdale had to push away. Also in the first period, Hill blocked Duff's dangerous cross and Miller was booked for a foul on Passley.
The early stages of the second period of extra-time were uneventful and Baxter curled a free-kick over the crossbar. On 115 minutes Fulham's best chance arrived when Duff and Sidwell combined before Kasami drilled just wide of the top corner.
Late on another Kasami shot was pushed around a post by Howard before the drama unfolded. In front of Blades fans, a Baxter corner from right saw Maguire win the first header before MILLER swooped to touch the ball home.
It was the very final minute of extra-time and Fulham had no time to mount a response as the Blades fans celebrated.
Fulham: Stockdale, Passley, Hangeland, Burn, Amorebieta, Kasami, Tankovic, Parker (Sidwell 69), Kacaniklic (Duff 59), Dempsey, Rodallega (Dejagah 59).
Unused: Stekelenburg, Karagounis, David, Christensen.
Blades: Howard, Harris (Hill 90), Brayford, Maguire, Flynn, Porter (Baxter 75), McGinn, Collins, Coady, Scougall (Miller 97), Murphy.
Unused: Long, Kennedy, De Girolamo, Dimaio.