Jurgen Klopp's side were looking to bounce back from their 3-0 home defeat to Atalanta BC in the Europa League on Thursday, but they suffered another loss in front of their own supporters.
Eze came up with the winner in the 14th minute, and the result has left Liverpool in third spot in the Premier League table, two points behind the leaders Manchester City, while Palace have moved into 14th, now eight points clear of the relegation zone.
Palace had a half-chance inside the first minute, with Eze finding space in a dangerous area, but his shot was gathered by Alisson Becker, who had returned between the sticks for Liverpool after a spell out through injury.
As expected, Liverpool started to gain control of the possession in the spell that followed, but Palace were dangerous on the counter-attack due to their pace and movement.
Darwin Nunez tested the reactions of Palace goalkeeper Dean Henderson with a strike in the eighth minute, but Palace silenced Anfield with the opening goal in the 14th minute.
Eze was the goalscorer, as the attacker tapped a low cross from Tyrick Mitchell into the back of the net from close range, with Liverpool's defenders nowhere to be seen.
Palace came incredibly close to doubling their lead in the 18th minute through Jean-Philippe Mateta, who raced clear before poking the ball past Alisson and towards goal, but Andrew Robertson just about managed to clear the ball off the line to prevent a second from going in.
Liverpool had a shout for a penalty in the 24th minute when Luis Diaz went down inside the box under a challenge from Daniel Munoz, but the referee was not interested and VAR agreed.
The home side came close to levelling the scores in the 27th minute when Palace failed to deal with a corner and the ball dropped for Wataru Endo, but the midfielder's effort came off the crossbar.
Palace goalkeeper Henderson then made a brilliant save to keep out a flying volley from Diaz after a dangerous cross from Robertson, who was carrying the fight for the home side, before Mohamed Salah was denied by the former Manchester United stopper.
The visitors continued to look threatening on the break, though, with Eze finding Michael Olise in a dangerous position late in the half, only for the latter to take a heavy touch at the vital moment.
Alisson had to get down to keep out a strike from Olise in the 42nd minute, before Mateta headed a Mitchell cross over from close range, with Liverpool wobbling defensively at the end of the first period.
Henderson made a wonderful save to deny Nunez early in the second period, with the Englishman standing strong to deny the Uruguay international, who just had to score from close range.
Trent Alexander-Arnold then fired over the Palace goal from just outside the penalty box after being set up by Salah, with Liverpool putting the pressure on at Anfield.
Virgil van Dijk headed a corner wide of the Palace target in the 70th minute, before Diogo Jota incredibly had a goal-bound effort blocked by Nathaniel Clyne, with the former Liverpool defender preventing Klopp's side from levelling the scores.
Palace had an incredible chance of their own in the 74th minute through Mateta, but the striker's effort from close range was brilliantly kept out by Alisson, before Curtis Jones placed one wide of the target down the other end one minute later after breaking into a one-on-one situation with Henderson.
Palace's goalkeeper made another smart save to deny Harvey Elliott in the 86th minute, and as expected, Liverpool were on the front foot, as the contest headed into seven minutes of additional time.
Salah had one cleared off the line by Mitchell in the first of those, and Palace held on to secure a famous three points, denting Liverpool's title hopes in the process.