Liverpool made one change to the side that beat Manchester City before the international break as Mohamed Salah failed to recover from an ankle injury and so dropped to the bench, with Alex Oxlade-Chamberlain handed a start.
Palace opted to keep things tight defensively in the opening exchanges as Liverpool dominated the majority of possession.
However, it was the hosts who had the first chance of the game on 16 minutes as Cheikhou Kouyate collected a poor clearance from Trent Alexander-Arnold and picked out the unmarked Jordan Ayew, who managed to fire wide seven yards from goal.
Liverpool continued to struggle in their quest to make a breakthrough, with just one blocked shot in the opening half hour, while Palace had five attempts at Alisson’s goal.
Palace thought they had taken a deserved lead on 42 minutes when Georginio Wijnaldum could only flick on a Luka Milivojevic free-kick towards the back post and James Tomkins was on hand to head in from close range.
But a push on Dejan Lovren by Ayew in the build-up was spotted by the Video Assistant Referee (VAR) and the goal was subsequently chalked off to the annoyance of the home players and fans.
Liverpool were brighter after the interval and Sadio Mane probably should have scored when he was played through by Jordan Henderson, but he put his effort half a yard wide of the post.
Mane was not to be denied a goal on 49 minutes after he collected a pass from Andrew Robertson in a tight area and managed to get a shot off from 15 yards, which took a deflection off Gary Cahill and went in via both posts.
Palace tried to respond on 53 minutes but Alisson tipped an Andros Townsend thunderbolt over his crossbar.
On 62 minutes, Mane guided a stooping header just wide, with Lovren inches away from touching the ball into the net at the back post.
Vicente Guaita then pulled off a fine save to keep Palace in the game minutes later when he tipped a Roberto Firmino right-foot effort wide of his bottom corner.
Palace looked to have rescued a point on 82 minutes when Wilfried Zaha was teed up by Townsend and slammed a 15-yard strike into Alisson’s bottom left corner of the net.
However, Liverpool went back in front with five minutes remaining as Palace failed to clear an Alexander-Arnold corner and the ball fell kindly to Firmino, who put his effort into the corner.
Palace almost snatched a stoppage-time equaliser but Zaha blazed his effort over the crossbar with just Alisson to beat from 10 yards.
The win makes it 30 games undefeated for Liverpool as they move on to 37 points, while Palace slip down to 13th on 15 points.
Crystal Palace 1-2 Liverpool
Liverpool made one change to the side that beat Manchester City before the international break as Mohamed Salah failed to recover from an ankle injury and so dropped to the bench, with Alex Oxlade-Chamberlain handed a start.
Palace opted to keep things tight defensively in the opening exchanges as Liverpool dominated the majority of possession.
However, it was the hosts who had the first chance of the game on 16 minutes as Cheikhou Kouyate collected a poor clearance from Trent Alexander-Arnold and picked out the unmarked Jordan Ayew, who managed to fire wide seven yards from goal.
Liverpool continued to struggle in their quest to make a breakthrough, with just one blocked shot in the opening half hour, while Palace had five attempts at Alisson’s goal.
Palace thought they had taken a deserved lead on 42 minutes when Georginio Wijnaldum could only flick on a Luka Milivojevic free-kick towards the back post and James Tomkins was on hand to head in from close range.
But a push on Dejan Lovren by Ayew in the build-up was spotted by the Video Assistant Referee (VAR) and the goal was subsequently chalked off to the annoyance of the home players and fans.
Liverpool were brighter after the interval and Sadio Mane probably should have scored when he was played through by Jordan Henderson, but he put his effort half a yard wide of the post.
Mane was not to be denied a goal on 49 minutes after he collected a pass from Andrew Robertson in a tight area and managed to get a shot off from 15 yards, which took a deflection off Gary Cahill and went in via both posts.
Palace tried to respond on 53 minutes but Alisson tipped an Andros Townsend thunderbolt over his crossbar.
On 62 minutes, Mane guided a stooping header just wide, with Lovren inches away from touching the ball into the net at the back post.
Vicente Guaita then pulled off a fine save to keep Palace in the game minutes later when he tipped a Roberto Firmino right-foot effort wide of his bottom corner.
Palace looked to have rescued a point on 82 minutes when Wilfried Zaha was teed up by Townsend and slammed a 15-yard strike into Alisson’s bottom left corner of the net.
However, Liverpool went back in front with five minutes remaining as Palace failed to clear an Alexander-Arnold corner and the ball fell kindly to Firmino, who put his effort into the corner.
Palace almost snatched a stoppage-time equaliser but Zaha blazed his effort over the crossbar with just Alisson to beat from 10 yards.
The win makes it 30 games undefeated for Liverpool as they move on to 37 points, while Palace slip down to 13th on 15 points.