Both sides threatened to take the lead in an eventful first-half which was extended by 12 minutes due to a floodlight failure at Craven Cottage.
Manchester United were agonisingly close to going ahead after eight minutes when Mark Schwarzer pulled off a fine save to prevent a Brede Hangeland own goal before Patrice Evra's effort came back off the woodwork and Rooney's overhead kick was cleared off the line.
United were equally as close to taking the lead when Rooney's curling shot from 18-yards hit the post and another Hangeland mishap saw his backwards header come back off the frame of his own goal, while Bryan Ruiz saw a half volley tipped onto the woodwork by David De Gea at the other end.
Robin van Persie saw a header saved by Schwarzer after half-time before Chris Baird's long range shot forced the impressive De Gea into action again.
Baird later saw a header cleared off the line by Rafael, just moments before Rooney's 79th-minute winner as he raced into the box and curled a composed shot into the far corner.
Philippe Senderos almost equalised for Fulham in injury time when his header was cleared off the line by Van Persie as United hung on for a win which takes them ten points above second-placed Manchester City, who play Liverpool on Sunday.