Title rival Manchester City trimmed the gap at the summit to four points when it beat the Reds' 2-1 last week and a much-changed XI was 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 hosts Wolves on Tuesday (AEDT), 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.
Other Premier League results:
Burnley 2-1 Fulham
Crystal Palace 1-2 Watford
Cardiff City 0-0 Huddersfield
Chelsea 2-1 Newcastle United
Leicester 1-2 Southampton
West Ham 1-0 Arsenal