The Reds raced two goals in front inside 20 minutes in an away game against Everton for just the third time in the competition's history thanks to strikes from Jordan Henderson and Salah.
Demarai Gray pulled a goal back before half-time to improve the mood around the ground, which began emptying after the second of Liverpool's goals, but Salah gave Liverpool some breathing space with his second.
Diogo Jota blasted home a fourth to highlight the gulf between the clubs and add some gloss to a win that keeps Liverpool within one point of Manchester City and two behind leaders Chelsea, with both sides also victorious on Thursday (AEDT).
Salah twice went close and Joel Matip wasted a big chance as Liverpool made a fast start, but the visitor's pressure soon told as Henderson guided Andy Robertson's cutback past Jordan Pickford with a first-time shot.
That was Henderson's first Merseyside derby goal in his 18th appearance and he also set up Liverpool's second goal, the midfielder playing in Salah following a Ben Godfrey error for his team-mate to bend around Pickford from an angle.
Liverpool continued to dominate but was given something to think about when Gray took Richarlison's pass in his stride and slotted in Everton's first goal in a month with 38 minutes played.
That kept alive the prospect of a fifth successive league draw between the sides at Goodison, but Salah restored Liverpool's two-goal lead with another calm finish after Seamus Coleman failed to control a poor pass from Gray on halfway.
Jota brilliantly turned Allan and fired an unstoppable drive past Pickford to round off the scoring 11 minutes from time in a one-sided derby that will lead to more questions being asked of Liverpool legend Benitez.