Still missing forward Philippe Coutinho through a back injury, after Liverpool reportedly turned down a third offer from Barcelona for the Brazilian playmaker on Friday, the hosts struggled to break down a stubborn Palace.
The Reds missed Coutinho's creativity for large sections of Saturday's win, but with Mane's help manager Jurgen Klopp's side are 1-0-1.
It was all Liverpool in the second half, and they finally broke down Palace's defence when Dominic Solanke capitalised on Milivojevic's poor control to play the ball to Mané who made no mistake.
"I have full confidence we will get our points very soon".
"You always need a finisher, and Sadio is ready for this when he is through on goal", said Klopp after watching the player score his sixth goal in eight Premier League matches. "We had the ball a lot but we had to be more creative".
"It was a long time ago that we played there and that's our place, that's where we want to play, that's where we want to be, especially after these two first games". The goal meant Palace had to come out and attack more, but apart from one Sullay Kaikai blast that was palmed clear by Mignolet, Liverpool were untroubled.
"He is a really good player but it is how it is with specialist left full-backs, it is hard; you play or it is possible you are not in the squad. They understand things won't change in one day".
"Obviously my decision not to read the English press is one of the best I did in my life", said Klopp, when asked about whether the win at Anfield had answered critics. I know Liverpool let in three at Watford but they didn't lose and then they won in midweek, which was a great result.
The atmosphere was electric throughout in Liverpool's first match at Anfield in this new season.
That was as close as Palace came to becoming only the fourth team in history to win four successive league matches at Anfield as they failed to properly test a team whose defensive frailty has regularly been exposed.