The Eagles took the lead in the second half when substitute Jean-Philippe Mateta won and converted a penalty.
But after Jordan Ayew picked up a second yellow card for a foul to stop a Liverpool counter-attack, the Reds levelled the game almost immediately as Mohamed Salah's shot found the net via a deflection off Nathaniel Clyne.
Dom Smith was at Selhurst Park to rate the Liverpool players’ performances…
Had little to do on his return from injury, but when called upon to keep Jefferson Lerma out with an exceptional first-half save, he was able to deliver. Also denied Joachim Andersen late on.
Trent Alexander-Arnold 6
Even on an off-day, Liverpool's No66 played a few of his trademark raking passing across the pitch for team-mates on the left. Did, however, play a few sloppy short passes into touch too.
Jarell Quansah 7
Was calm and composed on the ball and deserved his second Premier League start of the season, but kicked Mateta, handing Palace their penalty.
Virgil van Dijk 7
Good in the air. Booked for giving away Palace's first penalty, but had the yellow card rescinded when the spot-kick was overturned.
Kostas Tsimikas 7
One of Liverpool's better players. Took a pop shot from range in the first half which was speculative to say the least.
Dominik Szoboszlai 7
Energetic throughout, and tried to get Liverpool moving by carrying the ball forward and delivering crosses into the area.
Wataru Endo 5
Quite ineffectual. Went down too easily under pressure from Will Hughes, leading to Palace's first penalty being overturned. Was hooked at half-time.
Ryan Gravenberch 6
Replaced by Cody Gakpo on 57 minutes, having made little impression on proceedings at Selhurst Park.
Mohamed Salah 8
Not his best display of what has been a magnificent start to the season, but capped it off with his 200th Liverpool goal, before assisting their late winner.
Darwin Nunez 6
Was lively as ever, but scraped wide with a couple of decent chances.
Luis Diaz 5
Barely had an influence here, and the ball did not stick when it arrived at his feet.
Joe Gomez (Endo, 45') 6
Came on and played well at right-back, allowing Alexander-Arnold to move into midfield.
Ibrahima Konate (Quansah, 57') 6
Settled into defence well and kept Palace largely at bay towards the end of the game.
Cody Gakpo (Gravenberch, 57') 6
Was involved in Liverpool’s late wave of attacks in stoppage time as they piled the pressure on the hosts.
Curtis Jones (Nunez, 74') 6
Lively when he came on.
Harvey Elliott (Szoboszlai 74') 8
Scored a stunning winner in stoppage time.
Not used: Kelleher, Doak, McConnell, Bradley