United, who are second-bottom in League One, struck through substitute Shaun Miller's header in the 119th minute at Craven Cottage to set up a fifth round home tie with Preston North End or Nottingham Forest who meet in a replay on Wednesday.
Fulham, who prop up the standings and are four points adrift of the safety zone, made wholesale changes to the team that slumped 3-0 at home to Southampton in the league on Saturday but the reshuffle did not pay off on a rainy night in west London.
United, who beat top flight Aston Villa 2-1 away in the last round, just about deserved their win over a lacklustre Fulham side whose new signings were ineligible to play because they were not at the club before the first match.
(Writing by Ken Ferris; Editing by Ian Ransom)
