The match on the South coast lacked excitement for long periods, but Liverpool dominated for much of the contest despite requiring Salah to convert from 12 yards.
Liverpool now sit seven points clear at the top of the standings, while Brighton remain in 13th place despite their three-match unbeaten run in the top flight coming to an end.
An uneventful start set the tone for the opening 45 minutes of the game and although Liverpool quickly began to dominate possession, they were being left frustrated in their attempts to make an impact in the final third.
A cross from Andrew Robertson almost gave Roberto Firmino the chance to convert from eight yards, but it was Brighton who created the first opportunity with Glenn Murray heading over after meeting a cross from Pascal Gross.
Liverpool soon regained control, but aside from a glancing header from Xherdan Shaqiri which went wide of the far post Jurgen Klopp would not have been left encouraged by his team's performance from an attacking perspective.
There was little change as the contest edged towards half time, and Klopp needed to find a solution after the restart with Brighton looking increasingly comfortable in handling Liverpool's star quartet.
However, the Merseyside giants improved after the break, and shortly after being denied by a smart stop from David Button Salah was able to earn his team a penalty with his trickery inside the area leading to a clumsy challenge from Gross.
Salah was soon back on his feet to send an emphatic effort past Button, who could not get near the ball despite correctly guessing that the Egyptian would fire to his left.
Liverpool quickly set about trying to score a second, but despite keeping Brighton on the back foot chances continued to be few and far between and it led to the hosts remaining in the contest.
Firmino had a 20-yard strike well blocked by Button, before Georginio Wijnaldum fired the ball marginally off target from the edge of the area with his left foot.
Salah also squandered an opening with two minutes remaining, with the club's top goalscorer scuffing a shot wide from 10 yards after good work from James Milner.
However, Liverpool were able to comfortably see out four minutes of added-on time to post a positive response after back-to-back defeats to Manchester City and Wolverhampton Wanderers.
BRIGHTON & HOVE ALBION (4-5-1): Button; Montoya, Duffy, Dunk, Bong; March (Knockaert 66'), Propper, Stephens, Gross (Kayal 79'), Locadia; Murray (Murray 66')
LIVERPOOL (4-2-3-1): Alisson; Alexander-Arnold, Fabinho, Van Dijk, Robertson; Henderson, Wijnaldum; Shaqiri (Milner 72'), Firmino, Mane (Keita 90'); Salah (Origi 94')