Palmer opened the scoring with an exquisite left-foot strike, before heading home from Jordan Pickford's to make it 2-0 inside the opening 20 minutes.
The 21-year-old then took advantage of Pickford's misplaced pass to lift a right-foot shot over the Everton goalkeeper to complete a perfect hat-trick.
Nicolas Jackson added a fourth on the stroke of half time, before Palmer dispatched a spot-kick to net his 20th league goal of the season, moving him to joint-top of the Premier League scoring charts alongside his former Manchester City teammate Erling Haaland.
The commanding victory extends Chelsea's unbeaten run to eight league matches, while the Toffees are now winless in nine competitive away games.
Everton received a setback prior to the contest after it was revealed that Dominic Calvert-Lewin was ruled out through injury.
Calvert-Lewin's replacement, Beto, spurned a golden opportunity to find the net in the 10th minute at Stamford Bridge.
The Portuguese forward somehow conspired to lift the ball over from close range from Seamus Coleman's driven cross.
Everton's bright start was quickly forgotten as Palmer stamped his and Chelsea's authority on the contest with three goals in less than 20 minutes.
First, Palmer produced a neat piece of skill to beat Jarrad Branwhaite and exchange a one-two with Nicolas Jackson, before picking out the bottom corner with a precise finish.
The England international briefly thought he had missed the chance to double the advantage when he failed to find the net from Noni Madueke's fierce ball across the face of goal.
Just moments later, Palmer showcased his goalscoring instincts to nod home from a rebound after Pickford had parried Nicolas Jackson's shot.
Palmer wrapped up his hat-trick just before the half-hour mark, latching onto Pickford's misplaced pass to send his long-range strike soaring over the goalkeeper and into the net.
Sean Dyche's side experienced more anguish when Beto's 35th-minute header was ruled out for an offside infringement.
Jackson then compounded Everton's disastrous first half, collecting Marc Cucurella's cross before swivelling and drilling the ball in at the near post.
Dyche opted to make a triple change at the break, introducing Jack Harrison, Andre Gomes and Nathan Patterson in a bid to limit the damage.
Unfortunately for the Everton boss, he could only watch on as referee Paul Tierney pointed to the spot after Palmer and Madueke went down under challenges in the box.
Palmer wrestled the ball off Madueke to assume responsibility and slot home his fourth and Chelsea's fifth goal of a memorable evening.
Chelsea's victory became even sweeter when academy product Alfie Gilchrist came off the bench to thrash home his first senior goal in the 90th minute.
Mauricio Pochettino's side stay in ninth place but move to within three points of Newcastle United in sixth position, while Everton are left hovering two points above the drop zone.
Chelsea will now turn their attention to Saturday's FA Cup semi-final clash with current holders Man City, with the chance to progress to a domestic cup final for the second time this season.