A penalty from Palmer and Alfie Gilchrist’s first goal for his boyhood club rounded off the scoring in the second half.
Despite an eight-game unbeaten Premier League run, Chelsea remain in ninth but are now just three points adrift of sixth-placed Newcastle with a game in hand to come.
And they will head into Saturday’s FA Cup semi-final against City confident they can end the holders’ quest for a second consecutive treble.
A demoralising defeat leaves Everton still perilously placed just two points above the relegation zone.
The Toffees appealed against a two-point penalty for breaking Premier League sustainability rules on Monday, having also been docked a further six points for another charge this season.
How Everton could do with the boost of recovering some of those lost points as the battle for survival looks set to go down to the wire.
Nottingham Forest visit Goodison Park in a huge relegation six-pointer on Sunday and Sean Dyche’s men will need to improve at both ends of the field to prolong their 70-year stay in the top flight.
The visitors were dealt a blow before kick-off as Dominic Calvert-Lewin missed out due to a hamstring injury.
His deputy Beto spurned a glorious chance to open the scoring when he somehow turned over Seamus Coleman’s cross from point-blank range.
Palmer for player of the year?
Palmer was not so forgiving at the other end as he made another case to be crowned as the Premier League’s player of the year.
City must regret letting the 21-year-old leave for what now looks like a bargain £40 million ($50 million) in September.
Palmer nutmegged Jarrad Branthwaite before exchanging a neat one-two with Jackson and curling home from the edge of the box to open the scoring.
With Enzo Fernandez absent due to injury, Mauricio Pochettino flanked Palmer and Jackson with Noni Madueke and Mykhailo Mudryk for the first time in his starting line-up.
Everton could not live with the sharpness of that front four and Palmer was left with an easy task to head in his second after Jordan Pickford denied Jackson from a Mudryk cross.
The England goalkeeper then had a moment to forget as he gifted possession to Palmer, who nonchalantly chipped his international teammate on his weaker right foot from midway inside the Everton half.
In contrast to Palmer, Jackson has had an inconsistent first season at Chelsea but produced an excellent touch and finish to fire in his 13th goal of the season on the stroke of half-time.
A routine night for Pochettino’s men was still not without drama when they were awarded a penalty on the hour mark.
Both Madueke and Jackson tried to take over penalty duties before they were forcibly removed by captain Conor Gallagher to hand Palmer the ball.
He duly made it nine out of nine successful spot-kicks this season to take his tally for the season in both Chelsea and City colours to 25.
Academy graduate Gilchrist had only been on the field a matter of seconds when he blasted in the sixth after Pickford parried Ben Chilwell’s effort.
© Agence France-Presse
photo twitter @the_Lawrenz