Virgil van Dijk's towering header from Andy Robertson's corner opened the scoring just eight minutes in.
Robertson was the creator again when Alex Oxlade-Chamberlain doubled Liverpool's lead with a calm finish.
However, Palace served warning of what was to come in the second half before the break as Alisson Becker had to make fine saves from Michael Olise and Jean-Philippe Mateta to keep the Eagles at bay.
The Brazilian spread himself brilliantly to deny Odsonne Edouard's backheel, while Conor Gallagher somehow failed to turn in Olise's dangerous cross in a blistering start to the second period from the hosts.
Palace finally got their reward with a brilliant move as Jeffrey Schlupp and Mateta teed up Edouard for a tap-in.
Alisson was needed once again to keep his side in the lead and turn Olise's inventive lob behind.
But Liverpool's two-goal cushion was restored in controversial circumstances a minute from time when a penalty was given after a lengthy VAR intervention for Vicente Guaita's block on Diogo Jota.
"When we scored, the game was there for us to go and try and get that draw. But the referee killed our legs today," said furious Palace boss Patrick Vieira.
Fabinho sent Guaita the wrong way from the spot to seal the points to Klopp's relief.
Arsenal failed to bounce back from their League Cup semi-final defeat to Liverpool in midweek as Mikel Arteta's men were frustrated by a Burnley side that had not played a league game since 2 January.
The Clarets have up to four games in hand on some of their relegation rivals and gave their hopes of survival a huge boost despite remaining bottom of the table.
Nick Pope kept the Gunners at bay despite enjoying over 75 percent possession and having 20 shots on goal.
"We tried in every different way but lacked the quality in the final third," said Arteta. "At the end, when we have a really low block, you need spark and creativity to win football matches and today we didn't do that."
Arsenal edge up to sixth but are two points behind fourth-placed Manchester United.
Tottenham also have the chance to move into the top four when they travel to Chelsea later on Sunday.
Leicester's hopes of European football next season were dealt a huge blow by conceding twice deep into stoppage time to lose to Spurs in midweek.
Brendan Rodgers' men were again guilty of giving up a lead as Brighton secured a third consecutive 1-1 draw.
Patson Daka fired home on the rebound after Robert Sanchez had saved from Harvey Barnes in the first minute of the second half.
But Brighton levelled eight minutes from time when Danny Welbeck headed home Neal Maupay's cross.