A goalless draw against Everton on Sunday had raised questions about Jurgen Klopp's side following the return of the Premier League earlier this month.
But the Liverpool manager made four changes to the team that struggled in the Merseyside derby, restoring Mohamed Salah, Andrew Robertson and Georginio Wijnaldum to the starting line-up in the process.
Should Pep Guardiola's side win at Stamford Bridge, Liverpool could finish the job with a victory in their next game, against defending champions City, on July 2.
"Imagine if [Anfield] would have been full today and all the people could have experienced it live," said Klopp after the match.
"I don't think the game could have been better because my boys played like everybody was in the stadium. The atmosphere on the pitch was incredible," he said.
Alexander-Arnold's stunning 23rd minute free kick beat the visitors' defensive wall and goalkeeper Wayne Hennessey before nestling in the top corner to give Liverpool a deserved lead on their way to a 23rd consecutive Anfield win in the Premier League.
Fabinho then added a third on 55 minutes with a long-range rocket of a strike from around 30 yards out that left Hennessey no chance.
Salah turned provider from Liverpool's defensive half for the fourth goal after 69 minutes, an incisive first-time through-ball playing in Mane to bend a shot into the corner of the net.
"It was great to win like that and it was a great performance," Salah said postmatch. "All the players are motivated to win the league and we are motivated to play the best football behind closed doors."
It was such a dominant performance from Klopp's side that Palace did not have a single touch in Liverpool's penalty area -- the first time that has occurred in the Premier League since Opta began collating such stats in 2008.
Liverpool have 86 points from 31 games with City 23 points behind ahead of their trip to Chelsea.
"I have to watch the game tomorrow because we play them one week later," Klopp said of the Chelsea-City clash. "That's being professional and doing my job."
Information from Reuters was used in this report.