Jurgen Klopp's world champion side racked up a 16th consecutive Premier League triumph, and 24th from 25 top-flight games this season, with Salah striking twice in the last 18 minutes.
Alex Oxlade-Chamberlain opened the scoring in the 47th minute against his former club, unleashing a low drive past Alex McCarthy, before Jordan Henderson doubled the lead on the hour mark.
Salah produced a sweet finish for his first in the 72nd minute and showed his composure again at the death to send the Reds 22 points clear of second-place Manchester City, which travels to Tottenham Hotspur on Monday (AEDT).
It took 20 minutes for Liverpool to carve out the game's first opportunity, Salah's effort deflecting narrowly wide off the outstretched leg of Jan Bednarek.
Alisson was alert to paw away Moussa Djenepo's fierce strike shortly before the half-hour mark, while at the other end McCarthy made a double stop to deny Virgil van Dijk's backheel and Roberto Firmino's follow-up.
Danny Ings, Shane Long and Nathan Redmond all had presentable opportunities towards the end of the first half, but failed to take advantage of uncharacteristic uncertainty in the Liverpool defence.
Ings's penalty appeal after the restart was waved away by referee Kevin Friend, allowing Liverpool to storm up the other end and forge ahead through Oxlade-Chamberlain's low 18-yard drive in the 47th minute.
Salah saw a goal ruled out for offside after 54 minutes, but the Reds pulled further clear when Henderson coolly lifted the ball past McCarthy from 12 yards after being picked out by Firmino's clever cutback.
Salah eventually got on the scoresheet, latching onto Henderson's fine pass and clipping over McCarthy.
He bundled in a fourth from close range in the 90th minute, Firmino supplying his third assist, to add further gloss to yet another win for Klopp's relentless side.
