Mane also set a Premier League record to become the first player to score in nine successive games against the same opponent.
Jurgen Klopp’s side go top of the table after the result while Palace slip to 13th position.
However, it was Palace who started brightly and struck the woodwork in the opening two minutes after goalkeeper Alisson tipped the ball onto the post.
Two minutes before the break, Liverpool broke the deadlock when Guaita could only parry Salah’s header into Mane’s path for a tap in.
Mane has now scored 13 goals in 14 appearances against the south London club.
Liverpool controlled the second half and doubled their lead after 78 minutes when Salah struck home from Konstantinos Tsimikas‘s cross.
Keita grabbed a third with a fine looping volley with a minute remaining to seal the points for Liverpool.