Mane turned the ball in from close range just before halftime as he claimed a Premier League record for the most consecutive goals against one club - his ninth in a row versus Palace. He has 10 in all competitions against the London club.
Salah and Naby Keita, with a superb volley, added late strikes to give the scoreline an emphatic look after the home fans had earlier been left aghast over forward Diogo Jota’s early contender for miss of the season.
Liverpool provisionally move top of the table with 13 points from their opening five matches, though they have played a game more than most of their title rivals, while Palace are in 13th place with one victory so far in this campaign.
Mane brought up his century of goals for the Reds in all competitions, the 18th player to do so, when Salah’s header was pushed into his path by Palace keeper Vicente Guaita to give him an easy finish from close range after 43 minutes.
The goal followed a shocking miss by Jota from a similar situation as Thiago Alcantara’s header was only parried to the forward by Guaita but, with the goal at his mercy, the Portugal international blazed over the bar from all of two yards.
Salah added a second to kill off the contest on 78 minutes as Virgil van Dijk flicked on a corner and the forward supplied a neat finish.
But there was still time for Keita to add a magnificent third when he volleyed into the net from the edge of the box as the clock ticked into injury-time.
Liverpool were largely in control of the contest, but there were some bright moments for the visitors, including early on when home goalkeeper Alisson palmed the ball onto the post in a desperate attempt to keep out Wilfried Zaha’s looping effort.
Palace also thought they should have had a penalty as Konstantinos Tsimikas appeared to trip Christian Benteke in the box as the former Liverpool forward attempted to reach a cross.
Additional reporting by Reuters.