The win was Liverpool's sixth in a row, putting them in fifth place with 62 points, one point behind rivals Manchester United who hold the last Champions League qualification spot, albeit with two games in hand.
Brentford, on the other hand, are in ninth place with 50 points.
Liverpool took the lead in the 13th minute when Virgil van Dijk headed Fabinho's cross back for Salah to score his 100th goal at Anfield. Brentford's Bryan Mbeumo thought he had equalised before half-time, but his goal was ruled offside by VAR.
Despite Liverpool's dominance, they failed to convert their chances to extend their lead, with both Darwin Nunez and Cody Gakpo missing good opportunities.
However, Brentford only managed one shot on target throughout the game, allowing Liverpool to secure a vital three points.