Liverpool have also failed to score at home since their 4-1 defeat at the hands of Manchester City last month.
The result means that Jurgen Klopp's side remain four points off the top four places while Fulham go level on points with 16th-placed Brighton.
"There were moments we could have had more of an impact on the game -- conceding that goal before half-time was a massive blow," Klopp told Sky Sports.
We tried to react and the boys showed that. In the end we don't score goals and that's a big problem."
He added: "We never had a momentum in the season. You can see it's still in the boys. They still have it all but at the moment can't show it. It would be a masterpiece to find out how we change that overnight."
Despite Klopp saying he wouldn't turn his attention away from the Premier League, Liverpool made seven changes for the game ahead of their Champions League clash against RB Leipzig on Wednesday.
Fulham took a deserved lead on the stroke of half time when Andy Robertson failed to clear a corner and Lemina stole the ball off Salah on the edge of the box and produced a low shot into the corner.
Substitute Sadio Mane also came close to scoring after 70 minutes but his looping header clipped the post.
Liverpool pushed for a late goal but Fulham held on to secure an important win in their bid to remain in the league.
"It's a big win. There's a lot of emotion and I am very proud. I am happy for the players and the team because they got nothing short of what they deserved," Fulham boss Scott Parker told the BBC.
"We deserved this result and we have deserved better results over the last few weeks," Parker added.