Two goals by Crystal Palace forward Christian Benteke, who Liverpool sold to their opponents earlier this season, cost the hosts the chance to put pressure on the Premier League's top two with a rare home defeat on Sunday.
Philippe Coutinho had put Liverpool ahead in the 24th minute with a direct free kick. The home side dominated but conceded an equaliser just before half-time when Benteke, sold by Juergen Klopp in August, scored from a cross by Yohan Cabaye. In the 74th minute the Belgian stooped to head in a corner and win the game.
Klopp's team stay third but are five points behind second-placed Tottenham having played two games more.
Palace, who have recently beaten Chelsea and Arsenal, moved up four places to 12th and should now be safe from relegation after a sixth win in eight games.
Relive the action on Standard Sport's live blog!
Can't see the live blog? Click here for the desktop version.
Liverpool: Mignolet, Clyne, Matip, Lovren, Milner, Wijnaldum, Lucas, Can, Firmino,Origi, Coutinho.
Subs: Karius, Gomez, Grujic, Moreno, Woodburn, Alexander-Arnold, Brewster
Crystal Palace: Hennessey, Ward, Tomkins, Kelly, Schlupp, Cabaye, Milivojevic, Townsend, Puncheon, Zaha, Christian Benteke.
Subs: Speroni, Van Aanholt, Flamini, Campbell, McArthur, Sako, Delaney.
Referee: Andre Marriner
WHAT THEY SAID
Jurgen Klopp: "When the ball is in the air [Christian Benteke] is the main threat because he is unbelievable and we felt it all the time in training when he was here.
"You have to avoid passes in [the penalty area] but not only for him; if you only concentrate on Christian then [Wilfried] Zaha, [Andros] Townsend are around you.
"They have one of the best offensive lines in the Premier League and we respect this."
Sam Allardyce: "It is a big week for us to see how many points we can get. We want to take pressure off ourselves and get as many points as we can this week.
"At the end of this week there will only be three games left ... we are capable of getting points out of these games this week, as we have proved in the last six or seven games.
"I hope the players believe, as I believe, that they are capable of getting points out of them."