It marked the first time the Reds had won away in the top-flight by a margin of seven goals and was their biggest league victory since they hammered Palace 9-0 at Anfield in 1989 when they won the old top-flight First Division title.
The result put the defending champions on 31 points after 14 games ahead of second-placed Tottenham Hotspur, who host Leicester City on Monday.
Japanese midfielder Takumi Minamino repaid Juergen Klopp's decision to start him instead of Mohamed Salah by scoring after just over two minutes and Sadio Mane then made it two when he twisted and fired home form the edge of the box in the 35th.
Firmino added a third just before the break, flicking home to finish off a glorious Liverpool counter-attack that swept the length of the field, and Jordan Henderson added a fourth seven minutes into the second half as Palace fell apart.
Firmino got his second with a brilliant chipped effort in the 68th and Salah came on to net with a header before superbly curling a shot into the top corner five minutes from time.