Edinson Cavani's stunning long-range goal looked to have given United all three points as 10,000 fans returned to Old Trafford for the first time in 14 months, but Joe Bryan's late header earned Fulham a share of the spoils.
Cavani, who has scored 9 goals in last 10 matches across all competitions, struck with a driven effort on 15 minutes after catching Fulham goalkeeper Alphonse Areola off his line.
But Bryan, who scored the goals that saw Fulham win the playoff final and earn promotion to the Premier League, equalised on 76 minutes to deny United a much-needed win and punish them for wasting a string of chances to add to their tally.
Bruno Fernandes went close for United three times with long-range efforts after Cavani netted while Ademola Lookman and Fabio Carvalho missed gilt-edged chances at the other end before Bryan struck against the run of play.
The result left United second on 71 points from 37 games ahead of their final league match of the season against Wolverhampton Wanderers on Saturday, having stretched their winless run to three games.
Fulham, who are at home to Newcastle United in their final match, stayed 18th on 28 points as they picked up only their second point in the last nine league games.