The 21-year-old Palmer underlined his England credentials with a classic left foot, header, right foot treble and then added a second-half penalty as Chelsea ran riot, Reuters reported.
Nicolas Jackson and substitute Alfie Gilchrist were also on target as Chelsea remain ninth on 47 points but with games in hand on all the teams above them.
“When we signed him (Palmer) he was a great talent,” Chelsea boss Mauricio Pochettino said. “He is doing fantastic; he is doing really well and he has adapted himself. The impact on the team was amazing from day one.”
It was a sobering night for Sean Dyche’s visitors, however, as the heavy defeat leaves Everton 16th on 27 points, two places and two points above the relegation zone.
“A horrid night, miles away from where we want to be,” said Dyche. “The hard yards, simple tackling, basic principles - all out the window and it was a massive reminder of where we are.”
Everton actually had the first good chance but Beto blasted over from point-blank range after 10 minutes and it was downhill from then on.
Palmer got things moving after 13 minutes when he nutmegged Jarrad Branthwaite, played a one-two with Jackson and instantly clipped in a shot from the edge of the box.
Five minutes later more intricate passing – albeit alongside tentative Everton defending – set up the second as Jordan Pickford parried a Jackson shot straight to Palmer, who nodded the ball into the empty net.