The Reds had been leapfrogged by Arsenal prior to their trip to Craven Cottage, but they quickly moved back level on points with the Gunners thanks to two stunners from Trent Alexander-Arnold and Ryan Gravenberch before a Diogo Jota strike.
Marco Silva's men had restored parity before the break through Timothy Castagne, but the 12th-placed hosts were found wanting in the second period, seeing them miss the chance to rise into the top half of the rankings as Liverpool quickly reignited their title charge.
On the back of Thursday's unsuccessful Europa League exploits against Atalanta BC, Klopp changed almost half of his starting lineup, notably removing Mohamed Salah and Alexis Mac Allister from the XI.
Harvey Elliott was one of the five new faces brought in to face his old club, who could not call upon Willian due to a fresh groin injury; Bobby Decordova-Reid came in as the hosts' solitary change.
Full-backs steal first-half spotlight
As was the case in Bergamo, where they started at lightning speed before fading away in the second half, Klopp's side were on top in the earliest exchanges and should have been a goal to the good with just four minutes gone, but Luis Diaz headed a gilt-edged chance wide from Alexander-Arnold's cross.
Barring Diaz's early opportunity, Liverpool failed to convert their periods of possession into anything significant in attack in the opening half hour, but that was until Joao Palhinha took out Jota in Alexander-Arnold range.
The England international had left a previous free kick for Elliott - who struck the wall - just a few moments before, but he unsurprisingly stepped up for the Reds' next set piece and curled an inch-perfect strike over the wall and into the top corner in the 31st minute.
Bernd Leno's reach was futile, but the German did do well to tip a fierce Cody Gakpo over the bar in the 34th minute, after which Fulham responded well to Alexander-Arnold's opener and got their reward just before the break.
With two minutes of injury time gone, a close-range Rodrigo Muniz header was blocked by Jarell Quansah, but Castagne arrived on the scene to calmly side-foot home the Cottagers' equaliser into the bottom corner.
Gravenberch stunner ends Liverpool's open play hoodoo
The smiles were wiped off the faces of the home fans just eight minutes into the second period, though, as Alex Iwobi's lackadaisical pass was cut out by Elliott, who fed Gravenberch with a crisp forward pass and witnessed the Dutchman curl a stunning strike into the net via the inside of the post.
Gravenberch's first Premier League goal also saw Liverpool end their dry spell from open play, having last scored in such circumstances against Manchester United on April 7, and just like London buses, two came at once for the Reds.
Enhancing his reputation as a clinical finisher, Jota latched onto Gakpo's pass after the Dutchman beat Tosin Adarabioyo all ends up, and after just about holding his run, the Portuguese's low strike seeped under Leno's body and into the far corner in minute 72.
The second 45 was a near-total breeze for Liverpool, but Alisson Becker still had one last job to do to ensure a comfortable end to the game, pushing Tom Cairney's drive out wide and gratefully witnessing Muniz crash the side netting on the follow-up.
The Reds have regained their confidence at a crucial time ahead of Wednesday's Merseyside derby with Everton at Goodison Park, while Fulham have six days off before hosting London rivals Crystal Palace on April 27.