The Reds, who had lost this home fixture in each of the previous three seasons, netted 17 minutes from time after a quick exchange of passes set Mane free.
Visiting keeper Wayne Hennessey made a number of stops, particularly late on, but Palace will feel that they perhaps could have taken something from the game as Christian Benteke wasted a big chance when the scores were level.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
STARTING XI: Mignolet; Gomez, Matip, Klavan, Robertson; Milner, Henderson, Wijnaldum; Mane, Firmino, Sturridge
SUBS: Karius, Lovren, Flanagan, Can, Salah, Solanke, Origi
STARTING XI: Hennessey; Fosu-Mensah, Dann, Tomkins; Ward, Milivojevic, Puncheon, Van Aanholt; Townsend, Loftus-Cheek, Benteke
SUBS: Speroni, Schlupp, Kelly, Cabaye, McArthur, Kaikai, Lokilo
Jurgen Klopp: "The only thing we know is it's a tough game. Probably both teams were not overly happy [with their opening weekend]. We were not really disappointed about our first game but it was how a first game can be. There are things to improve and it's always like this, but even then we could've won which I think everybody knows. For Crystal Palace it was a little bit different. They want to strike back I'm sure but, again, it's Anfield and we have to show this."
Frank de Boer: "Hopefully the negative impact of the 3-0 loss [to Huddersfield] is gone. It will be a difficult game at Liverpool but we will do our best. Liverpool have fantastic players. It will be a very strong team that they put out. They are very dangerous in attack. Jurgen Klopp has selected the players to capitalise on winning the ball in key areas. The strength of this club is in the fans being supportive of the team. They understand things won't change in one day."
© Getty Images
LIVERPOOL SUBS: Karius, Lovren, Flanagan, Can, Salah, Solanke, Origi
CRYSTAL PALACE SUBS: Speroni, Schlupp, Kelly, Cabaye, McArthur, Kaikai, Lokilo
© Getty Images