Substitute Shaun Miller popped up in the final minute of extra-time in Tuesday's fourth round replay at Craven Cottage to head home from close range as the League One strugglers claimed their second Premier League scalp in the competition this season after beating Aston Villa to progress.
United's prize is a home tie against either Preston North End or Nottingham Forest, but for Fulham, who have scored only once in the last five games in all competitions and are bottom of the Premier League, the misery continues.
And Meulensteen will face plenty of scrutiny over his team selection.
With Fulham head coach Meulensteen making wholesale changes to the side that lost 3-0 at home to Southampton on Saturday, it was perhaps unsurprising that it was the visitors who made the brightest start when Jamie Murphy whipped a shot wide from 20 yards that had David Stockdale worried.
Although Meulensteen's men started to see more of the ball, the best Fulham could offer in terms of a threat was a Pajtim Kasami header that sailed harmlessly wide.
Some lovely build-up play involving Scott Parker, Kasami and Hugo Rodallega gave Clint Dempsey a sight of goal, but his effort was a wild swipe that ended a promising team move in disappointing fashion.
Sheffield United are languishing in the League One drop zone, but went close after 35 minutes when Conor Coady's inch-perfect cross found Murphy and he flashed his header just wide.
But it was the Premier League outfit who had the best chance of the half in injury time when Dan Burn headed Kasami's corner wide from six yards out.
Nigel Clough's United side resumed confidently after the restart with Bob Harris curling a free-kick just wide of Stockdale's goal before shooting straight at the goalkeeper moments later.
The game badly needed a goal and almost got one through calamitous circumstances after 65 minutes when Damien Duff's cross was sliced towards his own goal by Neil Collins, but the defender was bailed out by Mark Howard, who made an excellent one-handed save.
Both sides looked short of confidence as extra-time was required. John Brayford almost made a telling contribution when he was given space to drive through the Fulham defence, but his fierce low effort produced a superb save by Stockdale, who clawed the ball away.
Kasami went close twice late on with a crisp shot that went wide and again when Howard tipped wide.
But United won it at the death through Miller, who stooped to head in from close range after Harry Maguire had headed back across goal.
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