Darwin Nunez put Liverpool ahead with a superb chip before Alexis Mac Allister poked in a second at the Gtech Community Stadium.
Salah, back from a hamstring problem sustained on Egypt duty at the Africa Cup of Nations, got on the scoresheet with a crisp finish.
Ivan Toney pulled a goal back for Brentford, but Cody Gakpo added the gloss for Liverpool with a late strike.
The victory temporarily sent Liverpool five points clear at the summit, but Arsenal's 5-0 victory over Burnley later in the day reduced the deficit to two points again.
The only sour points for Liverpool manager Jurgen Klopp from the win at the Gtech Community Stadium were more additions to a lengthening injury list.
Curtis Jones and Diogo Jota were taken off before half-time, while Nunez was substituted at the interval.
"We have to see how much it costs us - we don't know that yet," Klopp said of the injuries.
"Curtis got a knock on the lower part of his shin, or above the ankle. We will see what that means. It was what he felt, but he was not 100 per cent sure. That Curtis cannot play on tells you that it must be something because he would have played on at all costs.
"Diogo looks probably the worst; I didn't see it back but I heard the pictures didn't look great as well, so we have to see there.
"Darwin, we took off because he felt really a little [something] but that was obviously today enough to immediately push the brake and that's what we did, so we took him off."
-- Courtesy of BBC Sport