The Red's calamitous home form took another turn for the worse ahead of their weekend fixture at Wembley as a spirited Cottagers side took all three points away from Merseyside
The result now moves the Londoners level on points with their Anfield counterparts, as Martin Jol's team remain on course to equal their best-ever Premier League finish.
Fulham performed with an air of control for most of the game, and they took an early lead thanks to Alex Kacaniklic's toe-poked shot deflecting in off Skrtel.
The Reds had chances of their own, with Jonjo Shelvey, Maxi Rodriguez and Andy Carroll all going close for the hosts, but Dalglish's men rarely stamped their authority on a game which, to all intents and purposes, looked like somewhat of a distraction ahead of the FA Cup final on Saturday.
Kenny Dalglish made nine changes from the team that started in the 3-0 victory over Norwich, with Andy Carroll returning to lead the line after recovering from a minor groin injury.
Fulham boss Martin Jol was again absent from the dugout as he continues to recover from a chest infection. There were three changes to the team that lost on Saturday, with Danny Murphy, Stephen Kelly and Alexander Kacaniklic all returning to first team action.
Liverpool hardly had time to settle into the game before Fulham took the lead in the fifth minute, as John Arne Riise galloped down the left and launched a low, scything cross into the boss, and after Alex Kacaniklic got a touch to it, Skrtel could only watch as it rebounded off his chest and into the goal past the helpless Doni.
And the Reds looked to make an immediate response, as Skrtel headed narrowly wide from an Aurelio free kick.
The next 10 minutes passed without incident as Liverpool looked to exercise some degree of control in the game, with Aurelio attempting to pick out Carroll.
Pavel Pogrebnyak spurned the chance to make it 2-0 on 15 minutes after a chipped through ball from Dempsey found the Russian in the penalty area with only Doni to beat, but the striker's volley was well saved by the goalkeeper before it was scrambled to safety.
And Fulham continued to penetrate at will as the bare bones Liverpool side seemed indifferent to the growing confidence in the London side, with Kacaniklic causing trouble among the Reds defence.
Maxi Rodriguez attempted to breathe some life into the Liverpool side on 20 minutes as the Argentine fired a 25-yard shot high and wide.
The industry nearly paid dividends in the 24th minute as Carroll laid-off to Kuyt just outside the area, whose snapshot whistled wide of the post by a matter of inches.
Jonjo Shelvey had the chance to equalise after good hold-up play from Carroll, who found the young midfielder in space in the box, and his driven shot went past Schwarzer looking destined for the bottom corner, only for Brede Hangeland clear the ball off the line after cleverly getting back.
And Liverpool continued to create more chances, as a corner was headed back across goal by Carroll and rebounded narrowly over by Maxi, before the two combined once more with the ball bouncing out for a goal kick after good play by Shelvey.
Shelvey saw a driven shot shave the outside post with his weaker foot before a Carroll header forced Schwarzer into a low, instinctive save.
Liverpool brought Downing on after the break in the hope that the impressive Carroll could get some more service, and the move immediately brought Liverpool to life, as Shelvey fired over from 30 yards barely seconds after the restart.
Spearing was next to test Schwarzer as Dempsey slipped while in possession on the centre circle and the young midfielder advanced goal-wards before unleashing a low, fizzing shot towards the corner, but the Fulham keeper was equal to it.
But Fulham threatened with more good chances to extend their lead, as a Damien Duff corner was headed powerfully, but narrowly wide by Clint Dempsey, and barely a minute later Moussa Dembele threaded the ball into the box for an unmarked Duff, who couldn't get a toe to it before Doni gathered.
Fulham had Hangeland to thank yet again, though, for another goal-saving effort as Maxi burst into the Cottagers’ box following a long ball and the big Norwegian slid in at the last minute to deny the Argentine.
Substitute Kerim Frei nearly made it 2-0 to the visitors on 62 minutes as he jinked into the box and hammered a low shot goal-wards, which flashed off the foot of the post, before Riise tested Doni after a scintillating run.
And such was Fulham’s dominance that Kenny Dalglish decided to haul off Fabio Aurelio for the reliable Jose Enrique, with the Anfield crowd becoming ever more inaudible as their team’s chances continued to look grim.
With 15 minutes remaining, the Fulham players began to believe they really could acquire their first league win at Anfield, with Dalglish finally giving the crowd something to cheer about by bringing on young talent Raheem Sterling for the ineffective Dirk Kuyt.
Dempsey could, and really should have killed the game off on 78 minutes as Frei threaded the American through, with only Doni to beat, but the Italian pulled off a fantastic one-handed save to keep the score at 1-0.
A 92nd minute Jonjo Shelvey free-kick represented Liverpool's last real chance, but the midfielder blasted well over the crossbar.