The hosts carried an early threat, and Tyrick Mitchell smashed a dangerous delivery across the face of goal, which could easily have been turned in by someone from either team.
The wing-back’s next delivery was more deliberate, picking out Yeremy Pino, who smashed his effort against the crossbar.
The Eagles then thought they had the lead when Jorgen Strand Larsen applied the finish after Liverpool failed to deal with a long throw, but VAR confirmed the on-field decision of offside because of a touch from Daichi Kamada.
Bradley Barcola missed a big chance for Liverpool, skying a close-range volley at the back post following a nice cross from Dominik Szoboszlai.
Kamada then had a similar opportunity for Palace, and came a lot closer, forcing Alisson to palm his effort away brilliantly.
That save looked even more crucial a few moments later, as Alexis Mac Allister headed in Szoboszlai’s free-kick to give Andoni Iraola’s side the lead.
The Brazilian goalkeeper showed his class again before the break, superbly turning Adam Wharton’s glancing header around the post.
Ronald Araujo almost doubled the advantage in first-half stoppage time, but could only guide his point-blank effort straight at Dean Henderson.
The Eagles again looked lively at the start of the second period, but a fast break from Liverpool gave them a mountain to climb, as Barcola picked out Florian Wirtz in the middle for a relatively simple finish.
Palace managed to salvage a late draw against Hoffenheim in midweek, but they seldom threatened to get back into this contest, with Alisson once again coming out on top to deny a rare clear-cut chance for Pino.
Henderson made a sharp stop of his own to prevent Alexander Isak from making things worse, but the Swede still managed to get in on the action with an 89th-minute strike following a nice interchange with Wirtz.
Sage’s side did at least get on the scoresheet after Milos Kerkez was deemed to have fouled his opponent as he went up for a header, but Eddie Nketiah’s cool penalty in added time was no more than a consolation.
Liverpool have now won more points away from home (13) than any other side this season, moving above Arsenal in that respect.
Iraola is also unbeaten on the road in the Premier League since the start of the year, a 14-game sequence only previously achieved by Sir Alex Ferguson, Jose Mourinho, and Ole Gunnar Solskjaer.
Crystal Palace have now lost each of their last four league matches, their worst losing streak since an equivalent run in 2023.
Patrick Vieira was dismissed three games into that streak, and Sage’s job must now be in danger.
Flashscore Man of the Match: Alisson (Liverpool)
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