Jurgen Klopp's side created numerous chances against a resilient Palace before Mane pounced on a rebound at the back post in the first half.
Palace had started brightly - hitting the post twice in as many minutes as Liverpool failed to clear their lines in the box - but the hosts' persistence eventually paid off.
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.
Mane later said, "I am lucky to score so many goals against Palace. Most important thing was the three points. Right place, right time. I am lucky. And happy. Thank you to all the fans, club and staff. Without them I am nothing.
"I would love to play against Crystal Palace. I am lucky. It would be great to play them every single weekend. In my head I am always thinking it is possible. I will keep trying and trying and trying.
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!