Ruben Amorim's United were left to rue Bruno Fernandes' wayward first-half penalty as they were pegged back at Craven Cottage. Rodrigo Muniz's own-goal put United in front on the hour mark, but Emile Smith Rowe came off the bench to rescue a point for Fulham.
Amorim is seeking an upturn on a disastrous 15th-placed finish last season, but so far his new £200 million forward line of Matheus Cunha, Bryan Mbeumo and Benjamin Sesko, who again started on the bench, is yet to fire.
Despite an encouraging performance, United lost 1-0 to title-chasing Arsenal in their season opener last weekend and will feel another two points slipped away in west London.
Fulham, though, felt the visitors got the better of two controversial calls. United's penalty was awarded after a VAR review. Fernandes is normally reliable from the spot, but this time blasted high over the bar.