Minamino's opened the scoring with his first Premier League goal after just three minutes. Trent Alexander-Arnold crossed a ball into Mane which he squared to the Japan international who emphatically hit it into the net.
Crystal Palace had plenty of chances but couldn't convert any of them and were made to pay on 35 minutes when a loose ball found its way to Mane who struck it into the bottom left corner.
Andy Robertson and Firmino teamed up for Liverpool's third with the Brazil international starting the move high up the pitch. He passed to Robertson who carried it up before returning the pass to Firmino who calmly struck the ball past Vicente Guaita.
There was little chance of Guaita getting near Henderson's goal on 52 minutes. The Liverpool captain struck the ball with his first touch from 22 yards out into the top left corner of the goal.
Firmino got his second on 68 minutes after Liverpool launched a counter-attack from a Crystal Palace corner. He took a touch before scoring from a tight angle.
Salah, who started the game on the bench, headed the ball home from close range on 81 minutes before adding his second on 84 minutes.
"I thought we were ruthless today and took our chances which is always important," Henderson said after the game.
"We dominated the game for large periods but I still thought we were sloppy at times.
"It was coming I felt as over the last few games we have had chances and not taken them which has cost us at times."