Liverpool manager’s team created many chances over a resilient Palace before Mane pounced on a rebound at the back post in the first half.
Mohamed Salah doubled their lead with just over 10 minutes left when he smashed home a flick-on from a corner, before Naby Keita’s late curling strike made it 3-0.
Liverpool were wasteful in the first half as Diogo Jota fired over the bar from a few yards out in a glorious position, while Thiago and Jordan Henderson were also denied by keeper Vicente Guaita.
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Palace had began brightly striking the post twice in as many minutes as Liverpool failed to clear their lines in the box but the hosts’ persistence eventually paid off.
In the end it was a comfortable victory that capped a good week for Liverpool at Anfield, following their dramatic 3-2 win over AC Milan in the Champions League on Wednesday.
And it means they go top of the table with four wins from their opening five games, while visitors Palace remain outside the top 10.
Line-ups
Liverpool
Formation 4-3-3
1Alisson
7Milner 5Konaté 4van Dijk 21Tsimikas
14Henderson 3Fabinho 6Thiago Alcántara
11Salah 20Jota 10Mané
Substitutes
8Keïta 12Gomez 15Oxlade-Chamberlain 17Jones 18Minamino 26Robertson 27Origi 47Phillips 62Kelleher
Crystal Palace
Formation 4-3-3
13Guaita
2Ward 16Andersen 6Guéhi3Mitchell
23Gallagher 8Kouyaté 18McArthur
9J Ayew 20Benteke 11Zaha
Substitutes
1Butland 4Milivojevic 5Tomkins 7Olise12Hughes 17Clyne 22Édouard 34Kelly 44Riedewald