Jurgen Klopp’s side were punished for a poor start by Eze after 14 minutes and then failed to take a host of chances in a frustrating second half on Sunday.
After last week’s draw at Manchester United and the midweek Europa League implosion against Atalanta, Klopp’s hopes of a glorious send-off are rapidly fading.
Palace were the latest side to throw a spanner in the works as they claimed just their second win under Oliver Glasner and eased fears of getting sucked into a relegation battle.
Eze’s goal marked the 21st time they have fallen behind in 32 league games this season and, unlike on many previous occasions, this time there was no coming back.
They pressed hard but could not find a way through Palace’s stubborn rearguard, succumbing to their first home league defeat in 29 games.
Liverpool had welcomed back first-choice goalkeeper Alisson Becker after 10 weeks and also brought back Mohamed Salah, Luis Diaz, Andy Robertson and Conor Bradley following the Atalanta humbling.
Yet it appeared to do little good as the hosts made a sluggish start and Palace made the early running.
The Eagles were rewarded as they pieced together a patient passing move, ending with Tyrick Mitchell cutting the ball back for Eze to casually side-foot home.
Things almost immediately got worse for Liverpool as a Virgil van Dijk slip allowed Jean-Philippe Mateta to race clean through on goal.
The Frenchman calmly lifted over the advancing Alisson but Andy Robertson did superbly to race back and clear off the line.
Liverpool belatedly began to stir and were unlucky in a goalmouth scramble as Wataru Endo scooped a shot against the bar.
Diaz then forced a save from Dean Henderson as he met a Robertson cross with a flying volley and Ibrahima Konate headed over from the resulting corner.
Alexis Mac Allister had a free-kick blocked and Salah tested Henderson after racing onto a long ball and cutting inside.
Despite the more positive play, Liverpool remained vulnerable and Michael Olise shot at Alisson before Mateta headed over.
Liverpool suffered a serious blow in the opening moments of the second half when Bradley was forced off with an ankle injury. His removal led to the return of Trent Alexander-Arnold after 13 games on the sidelines.
Liverpool pressed on and forced a corner from which Darwin Nunez looked certain to equalise but Henderson blocked his powerful strike at point-blank range.
Klopp tried to freshen up his front line by sending on Diogo Jota and Cody Gakpo and the former spurned a good chance when he was teed up by Dominik Szoboszlai but shot against a defender.
Curtis Jones was also wasteful when put through as he skewed a shot wide with only the keeper to beat.
Yet Palace also suffered frustration as Mateta was remarkably denied at close range by Alisson.
Liverpool looked laboured as time ran out but Salah was denied a leveller in stoppage time when Mitchell blocked him two yards out.
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