Both sides spurned chances throughout but the England striker showed fine composure to take advantage of a Philippe Senderos mistake to net the winner with just 11 minutes to play.
Sir Alex Ferguson made six changes from the team that narrowly beat Southampton 2-1 in midweek with the likes of Antonio Valenica, Nani and Rio Ferdinand returning.
Fulham made just two changes with Chris Baird and Ashkan Dejagah coming in for Sascha Riether and Steve Sidwell. New signing Urby Emmanuelson started on the bench, while a hamstring injury prevented Dimitar Berbatov from facing his former team.
The game started lively, and United will struggle to understand how they failed to break the deadlock after seven minutes, with two impressive Mark Schwarzer saves and Patrice Evra hitting the bar with the goal gaping keeping the scores level.
It was David de Gea's turn to excel after 11 minutes, superbly tipping over a wonderful John Arne Riise volley from 30 yards. The young Spaniard was kept busy for the home side, as Bryan Ruiz’s neat volley needed to be tipped onto the post just moments later.
With the woodwork still shaking, United went up the other end and suffered similar heartache when the ball deflected high from a corner onto Schwarzer's bar, with Antonio Valencia unable to convert on the rebound.
There was a nine-minute pause to proceedings close to half-time when the floodlights inexplicably went out to the delight of the 25,670 fans who wasted no time capturing the rare moment on camera.
Both teams were a more cautious after the break, presumably their manager’s half-time team talk fresh in their ears.
Chances were few and far between but Chris Baird heated things up with a stinging drive on the hour that De Gea did well to smother.
Van Persie broke the offside trap a minute later and teed up Nani on the byline, whose shot from eight yards was well blocked by Baird, while De Gea was perhaps fortunate to see Ruiz unable to capitalise on the goalkeeper’s parried save with 20 minutes remaining.
United were by no means the team on top, with Johnny Evans needing to be alert to head away from underneath his own crossbar as the pressure rose, while Rafael prevented a certain goal by heading away from a corner on his own goalline, but the away side struck the decisive blow.
Senderos got his angles all wrong trying to block Van Persie's run but only succeeded in letting Rooney run onto a long ball, and the striker made no mistake by curling the ball around Aaron Hughes and into Schwarzer's net.
Van Persie cleared another header off the line as the game entered the final five minutes and Chicharito missed a one-on-one in a brilliant ending, but United held on to open up a 10-point gap between themselves and second-placed Manchester City at the top of the Premier League table