The visitors struggled to click into top gear at Craven Cottage as Scott Parker’s team clinched a deserved point.
Fulham flew out of the blocks at the cottage with the returning Alisson forced into early action to save from Ivan Cavaleiro and Ademola Lookman.
And despite being denied a penalty after Fabinho tangled with Cavaleiro, Parker’s side were not to be denied as Bobby Reid fired them ahead.
Liverpool had three bites at the cherry to clear a corner and Lookman calmly rolled in Reid to smash past Alisson.
The Reds only manage to rouse themselves in patches before the break as Sadio Mane headed their best chance over the top.
The expected Liverpool reaction never really materialised after the restart as Mo Salah’s long-range effort was stopped by Alphonse Areola.
Klopp’s team did inevitably carry the bigger threat at Fulham began to bed in for the final 20 minutes, with Jordan Henderson missing an excellent chance.
Roberto Firmino’s cute pass freed the Reds skipper behind the Fulham defence but Areola was on him in a flash to make a block.
But with pressure creeping up and up, Klopp’s visitors grabbed their chance to equalise from the penalty spot.
Aboubakar Kamara handled Gini Wijnaldum’s free kick inside the box and Salah kept his nerve to hammer home.
Curtis Jones also set up a dramatic come back in the closing minutes but Areola stood firm to secure an unexpected point for Fulham.
Full Time at Craven Cottage it’s Fulham 1 Liverpool 1