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 following van Dijk’s flicked header from a Konstantinos Tsimikas corner.
Keita grabbed a third with a fine looping volley with a minute remaining to seal the points for Liverpool.
“It is one of the most hard-fought 3-0 wins ever,” Klopp told Sky Sports after the game. “We had to give everything. Hard but fair challenges. Physicality was immense and we had to be ready.
“We scored wonderful goals and used our set-pieces. It is massive in games like this. These are the games you have to win, when you are not brilliant.
“We weren’t flying today. We had to grind out a proper performance. I liked it. When you look back you have these games when it was really fun. This is one of those important games.”