Coming off the bench in Saturday's early game for his first appearance since injuring a hamstring on duty with Egypt at the Africa Cup of Nations, Salah netted his side's third goal moments after making a brilliant assist for Alexis Mac Allister to make it 2-0.
Darwin Nunez had opened the scoring with a superbly-taken chip to cap a wonderful counter attack in the first half as Jurgen Klopp's league leaders ensured they would finish the weekend still on top.
Ivan Toney continued his goalscoring form since returning from his gambling ban to briefly give Brentford hope at 3-1.
It was a fleeting moment of optimism, quickly extinguished when substitute Cody Gakpo stroked home Liverpool's fourth in the closing minutes.
Klopp's first half was tainted by the loss of Curtis Jones and Diogo Jota to injury, the latter departing on a stretcher.
On in his place came Salah, and the Egyptian should have doubled his team's lead five minutes into the second half, streaking clear of the defence and running through on Flekken, only to badly misjudge the finish and toe the ball wide.
He would make up for it with a sublime assist for Liverpool's second. Salah's arching ball into the box landed perfectly on the left foot of Mac Allister, closely attended by Kristoffer Ajer.
With one touch he brought the ball down and inside the defender, who skidded to the ground outmanoeuvred by Mac Allister's control, then with an athletic reach guided it with his toe into the bottom corner.
Salah's goal to put the result beyond doubt was simple as can be, Gakpo beating Ajer in the air. Salah ran onto the ball, and his physicality and touch were too much for Nathan Collins who couldn't stop him scoring.
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