Arne Slot’s side had already lost three times to Crystal Palace earlier in the season, but they started with a confidence belying such an unwelcome record.
They thought they had a penalty midway through the first half after Florian Wirtz released Mohamed Salah, who was felled by a last-ditch Brennan Johnson challenge.
However, a VAR intervention rightly gave the Eagles a reprieve, with the Wales international deemed to have got a sufficient touch on the ball. But the breakthrough seemed like it was coming, and Alexander Isak was ultimately the one to deliver it, bringing a speculative Alexis Mac Allister effort under his spell and diverting it beyond Dean Henderson.
Crystal Palace mustered a lively response, and Jean-Philippe Mateta tested Freddie Woodman at the near post after Liverpool had turned the ball over cheaply. From the resulting corner, the goalkeeper made an even better save, this time to deny Maxence Lacroix.
And the third-choice stopper was soon being serenaded by the Kop, after the Reds scored from the subsequent breakaway, a flowing move that saw Wirtz and Curtis Jones combine to set up Andy Robertson for what may be his last goal for the club.
Isak nearly added a second to his tally after the break, but Jaydee Canvot did really well to track him all the way, just about keeping Oliver Glasner’s side in the contest. Ismaïla Sarr came close to halving the deficit, but Virgil van Dijk did enough to prevent the forward from netting an eighth career goal against Liverpool.
However, Palace went on to grab a lifeline in bizarre fashion a little later, with Daniel Munoz chipping into an empty net after Woodman had injured himself making a smart save against Sarr.
Fourth-choice Liverpool keeper Armin Pecsi was getting ready on the bench, but Woodman was able to continue, and watched on in relief as Jorgen Strand Larsen could only find the post with the last big chance to equalise.
Wirtz then settled the issue once and for all, crowning a superb display with an excellent stoppage-time strike that flew into the far corner.
The one major downside for Slot’s side was the fate of Salah, who was forced off with an apparent muscle injury that may threaten to bring a premature end to his legendary Anfield career.
That thought certainly seemed to be on his mind, as he applauded all four corners of the ground before making his way off the pitch.
Palace have now won just one of their last 19 meetings against teams who started the day in the top half of the PL table – with the lone victory ironically coming versus Liverpool in September.
Meanwhile, the Reds took full advantage of Aston Villa’s lunchtime defeat and have moved up to fourth place on goal difference, with a healthy eight-point buffer to Brighton in sixth.