The Reds found themselves trailing at halftime after Ferdi Kadioglu's impressive strike put Brighton ahead.
However, two rapid goals from Cody Gakpo and Salah capitalized on Brighton's failure to convert their first-half dominance into a larger lead.
This marks the 13th win in 15 matches under Arne Slot, allowing Liverpool to move two points ahead of Manchester City, who suffered a 2-1 defeat against Bournemouth.
Both teams made significant changes from their League Cup encounter on Wednesday, which Liverpool narrowly won 3-2.
Despite the return of key players like Salah, Virgil van Dijk, and Trent Alexander-Arnold, Brighton controlled the first half and could have extended their lead before halftime.
Liverpool's best chance in the opening 45 minutes came when Darwin Nunez made a powerful run from his own half into the Brighton box, only to have his shot expertly saved by Bart Verbruggen.
Kadioglu struck first in the 14th minute with a powerful right-footed shot that hit the inside of the post.
Although Liverpool was again without their first-choice goalkeeper Alisson Becker due to injury, his replacement, Caoimhin Kelleher, made a crucial save to keep Brighton from increasing their advantage.
Jack Hinshelwood’s sublime pass split the Liverpool defence to set Georginio Rutter in behind Virgil van Dijk, but the Frenchman failed to beat Kelleher.
Kadioglu blazed over another big chance to make it 2-0 before a Danny Welbeck free-kick hit the side-netting.
Liverpool were forced into a change at the interval as Ibrahima Konate hobbled off with his arm in a sling and was replaced by Joe Gomez.
The England defender should have made an instant impact when he headed straight at Verbruggen when unmarked from a free-kick.
Slot’s half-time team talk had the desired effect as Liverpool tore out of the blocks a side transformed in the second half.
Alexis Mac Allister’s header was turned behind at his near post by Verbruggen before Van Dijk did not connect when picked out by Alexander-Arnold’s cross.
Brighton had kept Salah quiet for an hour but the Egyptian should have punished the visitors when he sped onto Nunez’s flick and failed to beat Verbruggen one-on-one.
The home side needed some fortune to get the comeback started as Gakpo’s attempted cross flew into the far corner without getting a touch.
With Anfield now in raptures it took just three minutes for Slot’s men to find the winner.
From a Rutter fluffed shot at one end, Liverpool broke at speed as Curtis Jones found Salah, who cut inside onto his trusted left foot and smashed into the top corner.
Brighton pushed the hosts back in a nervy finale, but Liverpool held on for a vital three points on a day that title rivals City and Arsenal, who were beaten 1-0 at Newcastle, both stumbled.