Goals from Alexis Mac Allister, Florian Wirtz and Alexander Isak secured all three points for the Reds, while Eddie Nketiah’s stoppage-time penalty offered Palace little more than consolation.
The victory was particularly important for Liverpool, who arrived in south London following consecutive defeats against Arsenal in the Premier League and Fenerbahçe in the Champions League.
For Crystal Palace, however, the defeat deepened an already worrying crisis. Four consecutive league losses have left the Eagles bottom of the Premier League table, increasing the pressure on manager Pierre Sage.
Liverpool started positively, with Mac Allister testing the Palace goalkeeper inside the opening 30 seconds.
But Palace gradually grew into the contest and demonstrated that they were not prepared to surrender their home advantage easily.
Daichi Kamada came agonisingly close to opening the scoring when his effort struck the crossbar in the 18th minute.
The hosts continued to threaten through Yeremy Pino and Jørgen Strand Larsen, while Liverpool goalkeeper Alisson Becker was forced into several important interventions.
Palace even thought they had taken the lead through Strand Larsen, only for the effort to be ruled out for offside.
Liverpool also missed opportunities, with Bradley Barcola failing to convert a promising chance after meeting a Dominik Szoboszlai delivery.
It was becoming a frustrating afternoon for the visitors, but their patience eventually paid off.
With five minutes remaining before half-time, Liverpool finally found the breakthrough.
Szoboszlai delivered a dangerous free-kick into the Palace penalty area, and Mac Allister timed his movement perfectly to meet the ball with a well-directed header.
The Argentine midfielder sent the ball into the net in the 40th minute, giving Liverpool a valuable 1-0 advantage.
It was a significant moment in a match where Palace had created several dangerous opportunities but lacked the finishing touch.
Mac Allister’s goal also highlighted Liverpool’s ability to make the most of set-piece situations, particularly when opponents were beginning to gain confidence.
The visitors went into the break with the lead, despite Palace having enjoyed some promising spells.
Wirtz produces the moment of quality
Liverpool returned from the interval determined to establish greater control, and they needed just nine minutes to double their advantage.
The goal was a beautiful example of quick attacking football.
Bradley Barcola burst forward down the right flank during a rapid counterattack before delivering a low ball into the penalty area.
Florian Wirtz had timed his run perfectly. The German international met the pass first time, producing a composed finish that beat the goalkeeper and made it 2-0 in the 54th minute.
It was an especially important goal for Wirtz, who had endured a 12-game Premier League scoring drought.
His finish demonstrated the technical quality and composure that Liverpool expect from one of their most expensive attacking signings.
More importantly, it gave Liverpool breathing space and forced Palace to chase the game.
The home side continued looking for a response, but Liverpool’s defensive organisation and Alisson’s outstanding goalkeeping made the task increasingly difficult.
Alisson stands tall as Palace search for a lifeline
While Liverpool’s attacking players deserved recognition, Alisson Becker was arguably one of the biggest reasons the visitors left London with all three points.
The Brazilian goalkeeper produced a series of crucial saves, denying Palace opportunities through Kamada, Pino and other dangerous attacking movements.
One particularly important moment came late in the match when Pino’s effort was stopped by Alisson. The goalkeeper then quickly recovered to deny Quinten Timber’s follow-up attempt.
His performance prevented Palace from building momentum when Liverpool were still protecting their two-goal advantage.
According to Sky Sports’ match report, Alisson finished the contest with six saves.
For a Liverpool side that had conceded too many goals during its difficult recent run, this was an encouraging defensive performance.
Isak puts the result beyond doubt
With Palace committing more players forward in search of a comeback, Liverpool found opportunities to exploit the spaces left behind.
Their third goal arrived in the 89th minute.
Wirtz was involved again, linking up with the advancing Victor Muñoz, whose movement helped create the opportunity. The ball eventually reached Isak, who finished clinically to make it 3-0.
It was an important moment for the Swedish striker, whose goal capped a strong attacking performance from Liverpool.
The visitors appeared to have completely settled the contest, and the travelling supporters could finally celebrate a comfortable-looking away victory.
However, Palace had one final moment to make the scoreline more respectable.
Nketiah scores a late consolation
In the fifth minute of stoppage time, Crystal Palace were awarded a penalty following contact between Liverpool defender Milos Kerkez and substitute Dario Osorio inside the box.
Eddie Nketiah stepped forward and converted from the spot to make it 3-1.
There was little time remaining for Palace to mount a serious comeback, and the final whistle confirmed a deserved Liverpool victory.
Although the late penalty denied Liverpool a clean sheet, it did little to change the overall story of the afternoon.
The Reds had been more clinical, more effective in decisive moments and ultimately more capable of turning opportunities into goals.
