Wayne Rooney scored the only goal late in the first half, firing high into the net from close range after Jonny Evans teed him up.
But United were a long way from their fluent best, and Fulham should have had a chance to equalise from the penalty spot in the 90th minute when Michael Carrick tripped Danny Murphy inside the penalty box.
Michael Oliver saw no foul, and United left the pitch with two more points than they should have had, moving three clear of City. On such things can championships be decided.
Having seen City struggle to break down Fulham at the weekend, United produced a similarly lacklustre display that failed to convert possession and pass completion ratio into anything resembling a goalscoring opportunity.
Fulham barely touched the ball in the first 10 minutes, but actually created the better early chances once they hit their stride.
David De Gea did well to punch a high ball clear under pressure, before Moussa Dembele fired wide after gliding past Michael Carrick.
Clint Dempsey then tested De Gea into two decent saves as both United and the Old Trafford crowd struggled to assert themselves.
United were struggling, with Danny Welbeck's touch unusually ragged up front, but eventually the chances came.
Patrice Evra shinned an overhead kick wide, then Rafael shot straight at Mark Schwarzer from a tight angle.
The opening goal came from an error by Fulham's sturdy defensive lynchpin, Brede Hangeland. The Norwegian failed to cut out an Ashley Young cross from the left, and Jonny Evans pulled the ball back for Rooney to ram it into the roof of the net from close range.
It was the England striker's 17th goal in as many appearances - though Lionel Messi is on a run of 18 in eight for Barcelona.
United were slightly better after half-time, as Antonio Valencia and Young forced decent saves from Schwarzer - the latter with a handsome curling effort.
The Australian made two more good saves from a powerful Young volley and the England player's follow-up, before Hangeland blocked Ryan Giggs's goalbound follow-up.
But the champions appeared happy to keep hold of the ball, frustrating Fulham instead of going for the jugular.
Substitute Murphy sounded a warning when he lofted a 40-yard pass over the top for Dempsey, who was just unable to make contact with the ball.
Then came the penalty appeal, when Carrick clipped Murphy as the former Liverpool man - a long-time United nemesis - burst into the box.
Referee Oliver waved play on, and United escaped.
Sir Alex Ferguson's sides have made an art form of carving victory from unconvincing performances, and this display may only strengthen their conviction that they have what it takes to outlast City.
Not vintage United, but certainly classic United.