Jurgen Klopp's side had drawn four and lost one of its top-flight games on the road after a 2-0 win at Chelsea on 21 September (AEDT), but a return to winning ways away from Anfield was never in doubt after Takumi Minamino's opener after 125 seconds.
It followed that up with fine goals from Sadio Mane and Firmino before half-time, while Jordan Henderson, Firmino and a brace from substitute Salah sealed a stunning victory in the second period.
The dominant win stretched Liverpool's lead at the summit to six points, although Tottenham Hotspur could reduce that back to three with a win over Leicester City on Monday (AEDT).
Liverpool wasted little time asserting its authority on the game, Minamino lashing his first Premier League goal into Vicente Guaita's bottom-right corner from 10 yards after being teed up by Mane.
The Reds doubled their advantage against the run of play 10 minutes before the interval, Mane flashing a fine strike past Guaita from 15 yards after collecting Firmino's pass.
The Reds then capped a wonderful first half in the 44th minute, Firmino controlling Andy Robertson's cross before superbly flicking past a helpless Guaita with the outside of his foot.
Things went from bad to worse for Palace after the restart as Henderson grabbed his first goal of the season, the captain expertly whipping into the far corner from 22 yards after being set up by Trent Alexander-Arnold.
Firmino then added his second in the 68th minute, lifting over Guaita from Salah's incisive pass before the Egypt international added a sixth with a close-range header nine minutes from time.
Salah added further gloss to the scoreline three minutes later, curling into Guatia's top-right corner from outside the penalty area.