Title rivals Manchester City trimmed the gap at the summit to four points when they beat the Reds' 2-1 last week and a much-changed XI were then knocked out of the FA Cup by Wolves.
But Jurgen Klopp's men restored a seven-point advantage thanks to the excellent Salah, having bided their time in the face of an often-impeccable defensive display from Brighton.
City host Wolves on Monday, by which time Tottenham will have had the chance to conquer Manchester United and close to within six of Liverpool.
Fabinho was deployed as an emergency centre-back for the visitors and his raking pass allowed Andrew Robertson to deliver a teasing cross from the left, with Roberto Firmino unable to slide home.
Room for Liverpool's attacking players to operate between a compact Brighton defence and midfield was sparing, but Sadio Mane engineered some in the 27th minute for Trent Alexander-Arnold to whip into the box and Xherdan Shaqiri to glance just wide.
Jordan Henderson found space for Liverpool's next headed attempt as Lewis Dunk made the block.
Salah escaped the shackles to force a save from David Button early in the second half and a jinking dribble from the Egypt star moments later drew a clumsy foul from Pascal Gross.
Button guessed correctly but was beaten for power.
Fabinho's first significant piece of defensive work came before the hour when he blocked from Gross, while Alisson took no chances in pushing Jurgen Locadia's speculative attempt around the post.
Georginio Wijnaldum lashed just wide 10 minutes from time and Salah uncharacteristically missed from close range when set up by substitute James Milner, but his earlier intervention proved to be enough.
Wobble ends for Klopp's men
Losing football matches can sometimes become a habit as tricky to shake as the winning addiction Liverpool operated with during the second half of 2018. The cup loss at Wolves on the back of their reverse at City is set to be a footnote in the Reds' season, but it would only have been human to feel nerves when this game was goalless at the interval. But they can relax again, for now, in the knowledge that City lost twice in December when they played later in the matchweek than Liverpool.
Salah lifts Liverpool with second-half salvo
Brighton's faultless defensive organisation got them through the first half with relatively few worries. At that point, Salah decided to take matters into his own hands. In the period immediately after the break – either side of winning and netting the game-breaking penalty – the superstar forward dropped deep and roved, breaking lines and Albion's otherwise unwavering concentration. He elevated himself above everyone else on the field to get Liverpool's title charge back on track.
