Bryan Ruiz hit the post for the hosts, while Rooney, Patrice Evra and a wayward header from Brede Hangeland cannoned back off the frame of the goal at the opposite end of the field.
An eventful encounter then saw the floodlights fail at Craven Cottage, forcing the players from the field shortly before half-time.
It looked as though that would be the major talking point in the game, but Rooney stepped up for United on 79 minutes - stroking the ball into the bottom corner after leading a counter-attack.
Released from the Old Trafford cabbage patch, United stated their intent by selecting two orthodox wide-men and attacking from the first whistle.
With Fulham, even minus former Red Devils striker Dimitar Berbatov, far more of a threat than they were when the sides met in the FA Cup seven days ago, it made for a rip-roaring opening in which both keepers excelled and the woodwork at either end was rattled.
Robin van Persie's corner started it, bouncing off Hangeland and towards his own goal, forcing Mark Schwarzer into the first of a series of excellent saves. Evra reacted quickest but could only scoop the ball against the bar before Sascha Riether cleared Wayne Rooney's shot off the line.
De Gea then got fingertips to a Ruiz shot that thudded against a post before play switched down to the other end where Hangeland, after denying Nani with a superb last-ditch tackle, turned Van Persie's corner on to the bar.
Only then did a breathless contest slow in intensity, although a brilliant move involving Nani and Tom Cleverley created another opportunity for Rooney, which he stroked against a post.
Nani was certainly making an impression on only his fourth match back after two months out with a hamstring injury, only for his momentum to be halted as the stadium was plunged into darkness.
The players did return to complete the opening period, although Van Persie's shot, gathered at the second attempt by Schwarzer, was the only incident of note.
The floodlight incident seemed to take the zip out of the entire contest, with neither side coming close to matching the thrills they had produced before the impromptu break in proceedings.
For a while it appeared Fulham might claim a major scalp.
Rafael cleared off the line from Ruiz after Jonny Evans had been forced to head away from underneath his own bar.
Rooney had other ideas, though, and when Fulham failed to get a touch to De Gea's long kick downfield, the England man picked his spot and, in front of national coach Roy Hodgson, calmly found the bottom corner.
Fulham piled forward in search of a dramatic leveller, but they were unable to break United's stubborn resistance - with Van Persie clearing a Philippe Senderos header off the line in stoppage-time.