Liverpool's return of 12 points means they are almost certainly already assured of a place in the knockout phase, but are closing in on sealing direct progression to the last 16 via a top-eight finish in the 36-team league.
Liverpool moved to the top of the Premier League with a resounding 3-0 win over Bournemouth as Tottenham eased the pressure on Ange Postecoglou by beating Brentford 3-1 on Saturday.
A second consecutive defeat leaves Manchester United boss Erik ten Hag already under pressure just three games into the new season.
Liverpool join Manchester City, Arsenal and Brighton as the only sides with maximum points from two games of the new Premier League season.
Crystal Palace defender Daniel Munoz struck in the second half after a superb run by Luis Diaz to capitalise on a disappointing display from La Roja ahead of this summer's European Championship in Germany.
Jurgen Klopp was whipping up the Anfield crowd at every opportunity and his side finally opened the floodgates on 56 minutes.
Liverpool will make their 14th appearance in the League Cup showpiece against Chelsea at Wembley on February 25.
Liverpool have flown the parents of Colombian star Luis Diaz over to spend Christmas in England as they recover from the ordeal of being kidnapped in their homeland
Luis Diaz scored a fairytale second-half double to fire Colombia to a 2-1 upset over Brazil on Thursday
Liverpool striker Luis Diaz had a tearful reunion Tuesday with his father, freed last week after a 12-day kidnapping ordeal at the hands of Colombia's ELN guerrilla group.