An open first half saw the woodwork rattled four times, with Bryan Ruiz hitting the post for Fulham, while Patrice Evra, Fulham defender Brede Hangeland and Rooney all sent the ball against the frame of the hosts' goal.
A floodlight failure in the 43rd minute plunged Craven Cottage into darkness and forced the players off the pitch, but they returned to complete the first half after the lights came back on.
Rooney netted the winner in the 79th minute, haring down the inside-left channel and steering the ball into the bottom-right corner to send United well clear of second-place Manchester City, who host Liverpool on Sunday.
"It's always difficult coming here and they made it hard for us," said Rooney. "We hit the post and bar three times in the first half, but we had to keep believing. We know we can create chances.
"We are in a good postion of course, but we are not going to get carried away."
Earlier in the day, new signing Moussa Sissoko completed a second-half brace with a last-minute winner to give Newcastle United a 3-2 victory at home to Chelsea that prevented the visitors from making ground on the top two.
After Jonas Gutierrez put Newcastle ahead in the first half, Chelsea hit back through fine goals from Frank Lampard and Juan Mata to take a 2-1 lead, but Sissoko equalised in the 68th minute before driving home his late winner.
"I don't need to say anything because his performance said it all," said Newcastle manager Alan Pardew of Sissoko.
"I remember watching him for Toulouse 14 months ago, and I thought he was the one for us. We watched him and set our sights on him for a long time."
Defeat left Chelsea six points behind City and 16 points shy of Manchester United.
"There were too many mistakes," said Chelsea interim manager Rafael Benitez.
"I could see a red card that was not shown for Fabricio Coloccini. We had a chance and Demba Ba received a broken nose, it is bleeding."
Arsenal left it late to beat Stoke City 1-0 and move to within a point of the Champions League places.
Substitute Lukas Podolski settled the game in the 78th minute with a low free-kick that deflected off Geoff Cameron to wrong-foot goalkeeper Asmir Begovic.
At Goodison Park, Marouane Fellaini headed in an injury-time equaliser to earn Everton a 3-3 draw in a breathless game at home to Aston Villa, who once again failed to defend a lead and remain firmly entrenched in the bottom three.
A Christian Benteke brace and a goal from Gabriel Agbonlahor meant Villa were 3-1 up with 21 minutes to play, Victor Anichebe having replied for Everton, only for a second-half double from Fellaini to give Everton a share of the spoils.
"I thought we played well but didn't defend well," said Everton manager David Moyes.
"Great credit to the players as they never gave up and kept going. We found ourselves chasing the game all the time and that makes for frantic football."
Villa are now two points below fourth-bottom Reading, who won 2-1 at home to Sunderland to keep themselves above the bottom three.
Jimmy Kebe scored both goals for the hosts, heading in an 85th-minute winner after Craig Gardner had cancelled out his early opener with a 29th-minute penalty.
Southampton were denied a precious victory after conceding a last-minute equaliser in a 2-2 draw at Wigan Athletic.
Gary Caldwell's 25th-minute header put Wigan in front, with second-half goals from Rickie Lambert and Morgan Schneiderlin giving the visitors a lead they held until Shaun Maloney snatched a goal at the death.
Meanwhile, on-loan Liverpool striker Andy Carroll scored only his second goal of the season to give West Ham United a 1-0 win at home to Swansea City.
Adel Taarabt had a second-half penalty saved as Queens Park Rangers were held to a 0-0 draw by Norwich City that left them six points from safety and four points adrift at the foot of the table.
"That's how it goes. Adel was outstanding again," said QPR manager Harry Redknapp.
"It's just one of those things. If we score, then we win the game, there's no doubt." – Sapa-AFP.