Quality was at a premium in a contest where the woodwork was hit three times - twice for Jurgen Klopp's side and once for the hosts, who remain without a top-flight win in 2023.
Klopp promised changes from the side that were humiliated 5-2 by Real Madrid in the Champions League - four to be exact - and the Reds' first-half performance typified that of a side totally devoid of confidence.
Despite dominating possession in the early exchanges, Liverpool's only chance of note was a header from Diogo Jota off the post in the 23rd minute, although the angle was far too tight for the Portugal international to make anything more of the chance.
Just two minutes later, Marc Guehi should have drawn first blood for Palace but headed wide from a brilliant position following a free kick, and Liverpool were awarded a set piece of their own in a dangerous area in the 29th minute.
However, in a move that largely summed up Liverpool's season, Trent Alexander-Arnold's effort struck the head of Jordan Henderson on its way behind, and it was soon Palace's turn to waste a gilt-edged chance.
Alexander-Arnold was caught in possession on the edge of the box by Jeffrey Schlupp, who cut back for Jean-Philippe Mateta, but the Frenchman's first-time effort from only a few yards out struck the bar and sailed over the top.
Klopp wasted no time in freshening the team up again - sending on Harvey Elliott for the underwhelming Naby Keita at the break - and the woodwork would be rattled again in the 49th minute, as Mohamed Salah's curler struck the underside of the bar from just inside the box.
Neither keeper was stretched in the half-hour that followed, with the Selhurst Park crowd not exactly getting their money's worth as they watched attack after attack break down thanks to a dodgy final ball.
The Reds finally carved out an opening in the 81st minute, as Salah fed Cody Gakpo, but Vicente Guaita was quickly off his line to put off the Dutchman, whose final act of the game was to shoot wide before he was replaced by Stefan Bajcetic.
A late corner and free kick for Palace came to nothing, who protested a decision for an unspecified offence after the full-time whistle, and the hosts remain 12th in the rankings ahead of a trip to Aston Villa next weekend.
Meanwhile, a point for Liverpool was enough to see them rise into seventh place ahead of Brighton & Hove Albion, and they are next in action at home to Wolverhampton Wanderers on Wednesday.