Palmer scored with his left foot, head and right foot in the 13th, 19th and 29th minutes on Tuesday (AEST) to become the first Chelsea player to score in seven successive home games.
The Manchester City reject stroked home a superb first from just outside the box and nodded in the second from close range after Jordan Pickford could only parry Nicolas Jackson's shot.
His third was a wonderful 35-yard lob after Pickford needlessly gifted him possession.
Nicolas Jackson added a fourth a minute before halftime, and Palmer made it 5-0 from the penalty spot in the 64th.
Substitute Alfie Gilchrist's first goal for the club made it six in added time.
Alfie Gilchrist spins away in delirium after netting against Everton. (AP PHOTO)
Palmer's goals took his season's tally to 20 in the league, tied with Manchester City's Erling Haaland.
The result extended a depressing few days for Everton. Just hours before kick-off, the club lodged an appeal against a two-point deduction for a second breach of the Premier League financial rules.
The Premier League said they intend to deliver a verdict before the last round of matches on May 19 "for clarity and certainty for all clubs and fans."
Everton were docked two points last week for overspending in a three-year spell up to the end of the 2022-23 season. It was a second deduction this season for the Toffees, who had six points removed — reduced from 10 following an appeal — for breaching spending limits from 2019-20 to the 2021-22 season.
The Merseysiders sit in 16th place, only two points and two places above the relegation zone.
After running their unbeaten streak to eight, Chelsea are in ninth place, only one point behind West Ham United. Chelsea have two games in hand over the Hammers.
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