Enzo Maresca’s side kept their Champions League hopes alive thanks to goals from Enzo Fernandez, Palmer, and a mix-up at the back between Jarell Quansah and Virgil Van Dijk.
Palmer has come under criticism after failing to find the back of the net in 19 matches. His performances have also dipped, but the 22-year-old put on a masterclass.
The England international played a helping hand in the first goal.
He picked up a deft position that bamboozled Liverpool before releasing Pedro Neto, who found Fernandez for the opener in the third minute.
Chelsea won the midfield battles against the makeshift Liverpool starters and looked devastating on the counter-attack, with Romeo Lavia playing some magnificent passes between the lines.
Levi Colwill and Trevoh Chalobah were indomitable at the back, with Robert Sanchez playing calmly, forming the perfect platform for the Chelsea attackers to thrive.
In the second half, Palmer wriggled away from Kostas Tsimikas to fire a cross in front of the Liverpool goal.
It did not go in, but Van Dijk smashed a poor clearance off Quansah and into the back of the net to make it 2-0.
The Dutchman made up for his error with a powerful header from a corner to reduce the deficit.
Unfortunately, his defensive partner did not recover after the own goal and gave away a penalty in the dying embers of the game.
Up stepped Palmer, whose last penalty was saved by Leicester City goalkeeper Mads Hermansen in March. He made no mistake this time, firing beyond Alisson Becker.
Palmer scored his first goal since January, earning Chelsea a massive three points in the race for a Champions League slot.
The Blues are fifth with games against Newcastle United and Nottingham Forest on the horizon.
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