Palace thought they had taken the lead after 25 seconds, but Eddie Nketiah was offside when he finished from close range, and it was Liverpool who opened the scoring after eight minutes thanks to Diogo Jota.
The Reds were in control of proceedings in the first half and frustrated Oliver Glasner's men by restricting them to just one shot by the interval.
Alexis Mac Allister was taken off at half time for Dominik Szoboszlai, but the change made little difference to the game, and it was Liverpool who had the first real opportunity of the second half when Mohamed Salah's shot was saved by Dean Henderson from inside the box after Szoboszlai passed towards the Egyptian from the right.
Eberechi Eze tested goalkeeper Alisson Becker in the 70th minute with a fierce strike near the edge of the box after the Liverpool defence failed to clear a long ball played towards Jean-Philippe Mateta.
Alisson worryingly seemed to pick up a muscle injury in the 77th minute, and with Caoimhín Kelleher unavailable for the Reds, 23-year-old Vitezslav Jaros was brought on made his Premier League debut.
Liverpool came under pressure from Palace in the latter stages of the game, but ultimately managed to hold on for all three points, and the victory means the Reds will stay top of the Premier League over the upcoming international break.
As for Crystal Palace, they have now lost four, drawn three and won none of their seven top-flight matches, and have failed to take themselves out of the bottom three.
CRYSTAL PALACE VS. LIVERPOOL HIGHLIGHTS
Diogo Jota goal vs. Crystal Palace (8th min, Crystal Palace 0-1 Liverpool)
Kostas Tsimikas plays a pass to Cody Gakpo from a narrow and deep position insider Palace's half, and the Dutchman crosses from the left to Diogo Jota, who pokes the ball home from the middle of the penalty area.
Jota scores his second Premier League goal of the season!
MAN OF THE MATCH - VIRGIL VAN DIJK
Virgil van Dijk once again proved how important he was for Liverpool on Saturday afternoon, and the centre-back won six of his eight duels against Palace.
The Liverpool captain was also composed in possession, completing 99 of his 108 passes and ending the game with 92% pass accuracy.
CRYSTAL PALACE VS. LIVERPOOL STATS
Possession: Crystal Palace 32%-68% Liverpool
Shots: Crystal Palace 9-16 Liverpool
Shots on target: Crystal Palace 5-4 Liverpool
Corners: Crystal Palace 3-8 Liverpool
Fouls: Crystal Palace 7-15 Liverpool
BEST STATS
WHAT NEXT?
After the international break, Glasner's side will take on Nottingham Forest at the City Ground in the Premier League on October 21, before facing Tottenham Hotspur on October 27.
As for Liverpool, they will host Chelsea at Anfield on October 20 before taking on RB Leipzig on October 23, when they will hope to make it three wins from three matches in the revamped Champions League.