Fulham have proved a tough opponent for Jurgen Klopp's men in recent years - the Cottagers are now unbeaten in their last three away encounters
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and Liverpool had to come from behind twice to cancel two Aleksandar Mitrović's goal.
The Serbian striker famously attempted to score in the Premier League but showed no difficulty against the Reds, opening the scoring for the campaign in the 32nd minute.
Darwin Nunez canceled out Fulham's lead with a clever impromptu heel strike just after the penalty spot, before a penalty from Mitrovic restored the home side's advantage.
With less than 10 minutes to play, however, Nunez then set up Salah, who calmly scored the vital equalizer for Liverpool.
The warning signs were there for Klopp as Liverpool struggled to start and chained passes together early, before being caught at the break.
Although they have grown in the game, it is far from a sober performance from the Reds, while Fulham look completely different from a team
that has scored 10 goals in their opening three Premier League games in 2020-21.
It was not the start of the season Liverpool could have envisioned when the fixture was announced,
with Fulham applying most of the initial pressure and Mitrovic being pushed away in the second minute.
However, Liverpool had the ball in their net with the first real attack of the game, Luis Diaz's shot past Marek Rodak, but Andy Robertson was offside on the move.
Liverpool continued to struggle in the game and it was Fulham who unexpectedly took the lead in the Champions League final.
The Cottagers made a smart move on the left, with Kenny Tete past Neeskens Kebano before delivering a deep cross to the post, where Mitrovic beat Trent Alexander-Arnold
Liverpool nearly equalized the next minute - Diaz's effort from inside the box flew past Rodak but missed the post.
Klopp had plenty of injury problems before the game, but his luck took a turn for the worse when Thiago Alcantara came on early in the second half and had to be substituted,
Early in the second half - after a few quiet minutes - Kebano almost scored a second for Fulham when he hit the ball high over the post and from the corner that followed,
Bobby Decordova-Reid was saved by Alisson. There were signs of intent from Liverpool as Nunez searched for Diaz in the far corner, but his attempted pass was blocked by Palhinha.
After leveling, Liverpool immediately looked like they were on the cusp of victory, with Diaz having a shot saved and Tim Ream also having to come to the rescue.
Mitrovic came up to take the kick and Alisson guessed the right track but was defeated by the power of the shot.
Liverpool equalized once again when Alexander-Arnold crossed from the right to the center of the box, Nunez scurried away for Mohamed Salah to head in.