United, hoping to exert even more pressure on second-placed Manchester City who host Liverpool on Sunday, laboured at times but Rooney's 79th-minute strike proved decisive.
Both sides struck the woodwork in the first half, Patrice Evra for United and Brian Ruiz for a lively Fulham side, and Rooney also thumped a shot against the post in the second half before capitalising on an error to make the breakthrough.
Philippe Senderos, whose misjudgement allowed Rooney to race on to a long ball and score, nearly equalised in stoppage time with a header that was cleared off the line by Robin van Persie.
United have 62 points from 25 games with champions City on 52, having played one game less.
