Mario Lemina's goal on the stroke of half-time, after the Gabon midfielder had dispossessed Mohamed Salah, gave the visitors a deserved win and saw them draw level on 26 points with 17th-placed Brighton.
The Reds, who won the title last season by 18 points, had gone 68 home league matches unbeaten before their current run started with a loss to Burnley in January.
Liverpool boss Jurgen Klopp made seven changes following the defeat to Chelsea in attempt to freshen things up, but Fulham were the better side and carved out four chances in the opening half an hour alone.
Ademola Lookman, who was given the freedom of Anfield, twice got behind the Reds defence but could not finish while Josh Maja was denied by a brave save by Alisson.
Diogo Jota, making his first start since 28 November, was denied an equaliser by an acrobatic save from Alphonse Areola early in the second half before substitute Sadio Mane hit the post with a looping header.
But Liverpool were poor and have now failed to win their last eight home games.