A stoppage-time header by Darren Bent saw Fulham snatch what could prove to be a precious point at Old Trafford.
Fulham were the victims of an FA Cup upset on Tuesday as a last minute extra-time goal sent Sheffield United through.
Fulham slipped to the bottom of the table after a fourth successive league defeat.
Second-half goals from Jonjo Shelvey and Chico Flores earned Swansea City a 2-0 win over Fulham on Tuesday night.
Fulham and Sheffield United will go again in the FA Cup after drawing 1-1 at Bramall Lane.
Two goals by Santi Cazorla in five second-half minutes saw a plucky Fulham side fall to defeat to league leaders Arsenal.
Fulham eased into the Fourth Round of the FA Cup after a comfortable Third Round Replay win in SW6.
Fulham’s recent good form came to an end in a heavy home defeat to fellow strugglers Sunderland.
Fulham and Norwich City will meet again in the FA Cup with Budweiser after drawing 1-1 at Carrow Road.
Fulham came from behind to register a massive win against 10-man West Ham on New Year’s Day.