Salah — who now has 200 goals in English football with Liverpool and Chelsea — became the first player in history to score in each of Liverpool’s first six home league games of a season as Jurgen Klopp’s side climbed above Tottenham Hotspur and Arsenal into second in the table on 27 points after 12 games.
Brentford, who saw their three-match winning run end, are 10th on 16 points.
Darwin Nunez had two first-half efforts disallowed for offside as the sense of an impending goal grew.
Talisman Salah finally broke the deadlock in the 39th minute after Trent Alexander-Arnold drilled a pass to Nunez who slipped it to Salah behind Brentford’s defence for a left-footed finish into the far corner.
The 31-year-old Salah headed in his second at the back post in the 62nd minute after Kostas Tsimikas slid to keep the ball in play on the byline. Diogo Jota scored in the 74th when he cut inside and bent the ball inside the far corner past keeper Mark Flekken.