Trent Alexander-Arnold’s fantastic free kick into the top corner gave the Reds the lead in the 32nd minute, but Timothy Castagne’s header in first-half added time pulled the home team level.
However, the visitors took the lead again early in the second half thanks to a crisp strike from Ryan Gravenberch, and they secured the points with a signature Diogo Jota finish.
The outcome puts Liverpool and leaders Arsenal at 74 points apiece after 33 games, with the Gunners taking first place on goal differential. Manchester City is now in third place with 73 points after 32 games.
Palhinha’s reckless challenge denied Jota his driving run, and Alexander-Arnold curled a beautiful, unstoppable shot into the top left corner from the free-kick that was placed just under 25 yards out to the left of center.
Cody Gakpo’s well-hit shot was superbly saved by home goalkeeper Bernd Leno, as the Reds looked to build on their early lead.
But the visitors soon needed a goalkeeper of their own. Palhinha grabbed the ball from Endo and found his teammate, but Alisson was equal to Rodrigo Muniz’s close-range strike.
But after two minutes of added time, Fulham leveled the score. When Muniz attempted to head the ball in their attack, Quansah stopped it, but Castagne was there to finish through the bodies.
Elliott had an opportunity early on to give Liverpool the lead again. Jota set up the No. 19 well in the ‘D,’ but his effort was dragged off target.
However, Gravenberch’s second goal of the afternoon came quickly for Klopp’s team, and it was another beauty as Elliott, who had previously cut off Alex Iwobi’s ball from behind, released Gravenberch before curling a 20-yarder in off the right-hand post.
After Gravenberch and Robertson had broken through, Leno stopped Jota’s poke at goal from yards out, and the Cottagers’ keeper then denied Gakpo twice more.
But now that they were playing with some zip, the Reds ensured a vital win in the Premier League title match on 72 minutes.
After Gakpo’s through pass allowed Jota to come down on goal from an angle in the left channel, the Portuguese forward pushed a shot over Leno’s reach to guarantee that the traveling Kop behind that goal could celebrate their victory and leave Craven Cottage.