Daniel Sturridge equalised after Kolo Toure’s ninth-minute own goal had gifted Fulham the lead at Craven Cottage.
Rene Meulensteen’s side, bottom of the Premier League but buoyed by Sunday’s draw at Manchester United, had Liverpool on the back foot straightaway and Simon Mignolet saved Ryan Tunnicliffe’s low shot in the opening minute.
The Whites kept up the early pressure and Toure had to react quickly to prevent Darren Bent getting to Kieran Richardson’s left-wing cross.
But the next time Richardson crossed from the left Toure contrived to slice the ball into his own net from close range.
On-loan Tottenham midfielder Lewis Holtby almost doubled the lead with a 20-yard effort that fizzed just wide.
And Bent headed over from seven yards out after being found by another Richardson cross.
Having struggled for most of the first half, Liverpool levelled out of nothing four minutes before the interval.
Steven Gerrard carved open the Fulham defence with a perfect through-ball and Sturridge fired in off the far post.
Liverpool made a strong start to the second half but were given a scare when Mignolet had to back-peddle to gather Steve Sidwell’s cheeky 60-yard attempt to catch him off his line.
Luis Suarez miscued twice before the break but has looked increasingly dangerous since the restart, and on the hour mark he hit the post from the edge of the penalty area.
Fulham: Stekelenburg; Riether, Heitinga, Burn, Riise; Tunnicliffe, Sidwell, Kvist, Richardson; Holtby; Bent.
Subs: Stockdale, Hangeland, Kasami, Kačaniklić, Duff, Cole, Parker.