The Reds claimed the game’s only goal on 39 minutes when Mohamed Salah emphatically converted a penalty after Issa Diop had fouled Darwin Nunez inside the box.
Jürgen Klopp’s side had further chances to extend their advantage, while Alisson Becker thwarted Carlos Vinicius with a fine save in the closing stages to ensure the hosts extended their unbeaten run to seven matches.
Team news
Liverpool made three changes from the team deployed against Tottenham Hotspur on Sunday.
Nunez started in attack, while Jordan Henderson and Kostas Tsimikas also came back into the line-up. Harvey Elliott, Cody Gakpo and Andy Robertson were the ones to make way.
Liverpool: Alisson, Alexander-Arnold, Konate, Van Dijk, Tsimikas (Robertson, 65), Fabinho, Henderson, Jones (Milner, 83), Diaz (Gakpo, 65), Salah (Elliott, 83), Nunez (Jota, 83).
Unused subs: Kelleher, Gomez, Carvalho, Matip.
First half
Liverpool made an assertive start as they went in search of a fifth consecutive Premier League win.
With three minutes gone, a Nunez cross from the right was deflected invitingly into the path of Trent Alexander-Arnold just outside the box. He took aim, but his strike ricocheted off Luis Diaz and went wide.
Diaz himself then tried his luck at Bernd Leno’s goal, but dragged off target, while at the other end Virgil van Dijk had to produce a crucial interception to prevent Harry Wilson’s low ball picking out Vinicius for a tap-in.
Before the 10-minute mark, Alexander-Arnold threatened on two further occasions, first having a drive saved by Leno before sliding a low shot a fraction the wrong side of the post.
In what was proving a lively contest, Liverpool then required Alisson to keep out an edge-of-the-box hit from Vinicius with his fingertips after the forward had spun into space.
Next, Salah collected an Alexander-Arnold pass inside the area, shimmied beyond Diop but watched his scoop over Leno drop beyond the far post.
Diaz had a shot flick away from goal off a white shirt and Joao Palhinha’s near-post back-heel was kept out by Alisson as the teams continued to trade chances.
Then, with 38 minutes on the clock, Liverpool were awarded a penalty. Nunez closed down Diop on his blindside inside the area before going to ground as the defender swung at the ball to clear.
Salah assumed responsibility and emphatically hammered beyond Leno for his 29th goal of the season – and to give the hosts a half-time advantage.
Second half
Liverpool fashioned two good opportunities in the opening exchanges of the second period, with Nunez stretching to stab wide from a Henderson through ball and Salah striking over after fine work by Diaz to tee up the chance.
Gakpo and Robertson were introduced into the action with 65 minutes gone, replacing Diaz and Tsimikas, with the former quickly involved, having a close-range attempt smothered by Leno.
As the game edged towards its conclusion, a trademark save from Alisson denied Fulham an equaliser when he pushed a shot from point-blank range away to thwart Vinicius following a counter from the visitors.
Deep into stoppage time, Fulham substitutes Manor Solomon and Bobby De Cordova-Reid combined, but the latter swept over from 10 yards out to ensure Liverpool would pick up three points and continue their momentum.
Attendance: 52,602