League One outfit United, who beat Aston Villa 2-1 in the fourth round, looked to be on course for a second successive Premier League giant-killing when striker Chris Porter gave them the lead midway through the first half at Bramall Lane.
However, the game turned early in the second period when United captain Michael Doyle was sent-off for violent conduct.
Rene Meulensteen's side could not break through a determined home defence in the closing stages, with Darren Bent somehow shooting wide from point-blank range late on as United held on.
Fulham made a bright start to proceedings and created the game's first chance after five minutes as Rodallega headed narrowly over the crossbar.
However, the visitors were forced into a change shortly after, Ange-Freddy Plumain replacing John Arne Riise following the Norwegian's collision with an advertising hoarding.
Jose Baxter curled wide from a free-kick for Clough's men just after the 20-minute mark, as chances proved few and far between in the early stages.
But United took the lead with their first real opportunity of the game after as Porter turned home Harry Maguire's shot across the face of goal.
Still, that dismissal did not impact the hosts' ambition, and midfielder Ryan Flynn - who scored the winner at Villa Park in round three - came close to adding a second when his near-post shot was kept out by David Stockdale.
Fulham dictated possession in the second half as their numerical advantage began to tell, but Meulensteen's men were fortunate not to concede a penalty for Aaron Hughes' apparent trip on Flynn.
Again Marriner dismissed the claims, and Fulham's quality told in the 75th minute, Rodallega finding the bottom corner from the edge of the area with a fierce low finish.
Defender Philippe Senderos hit the bar with a header eight minutes from time before Bent's incredulous miss encapsulated a frustrating afternoon for Fulham.
