Palmer, who returned to the side following a one-match suspension, took full advantage of poor Luton clearance to give Mauricio Pochettino's side an early lead in the lunchtime kickoff.
Luton's task became much tougher when Noni Madueke thumped home Chelsea's second goal in the closing stages of the first period.
Although Palmer netted a second to extend the visitors' advantage, an undeterred Luton side managed to set up a nervy finish thanks to late efforts from ex-Chelsea midfielder Ross Barkley and substitute Elijah Adebayo.
Unfortunately for the hosts, they ultimately ran out of time, as the visitors saw out the remainder of the contest to claim their first away win since November 6.
After scoring the winner against Crystal Palace on Wednesday, Madueke was presented with his second league start of the season.
The 21-year-old carved out the first meaningful chance of the game, teeing up Conor Gallagher to fire straight at Thomas Kaminski.
The visitors ultimately found the breakthrough in the 12th minute after Luton failed to clear their lines following Nicolas Jackson's near-post effort.
Issa Kabore's weak clearance was gathered by the grateful Palmer, who quickly set the ball out of his feet to drill a fierce strike into the bottom corner.
Not content with their narrow lead, Chelsea managed to add to their tally before the half-time break thanks to Madueke's unstoppable strike.
The winger collected the ball from Palmer, before working the space to smash a right-foot shot past Kaminski at the near post.
Although Luton showed signs of improvement after the break, their failure to test Djordje Petrovic ultimately proved costly as Palmer put the game beyond doubt in the 70th minute.
The former Manchester City man demonstrated his composure in front of goal to round Kaminski and chop the ball onto his favoured left foot before tapping into an empty net.
After conceding a third, Luton thought they finally had a goal of their own to celebrate when Adebayo nodded in from Alfie Doughty's inviting cross, only for VAR to rule out the effort for an offside infringement.
Moments after having a goal chalked off, Adebayo experienced more frustration when he sent his headed effort crashing off the woodwork.
While Adebayo was left ruing his luck, his teammate Barkley had no such problems in finding the net, as the former Chelsea man flicked his header home from Doughty's corner to give the home faithful something to cheer about.
Luton had a second goal to celebrate in the 87th minute as Adebayo finally netted the goal his substitute performance deserved when he tapped home from close range after Petrovic had made a smart reaction save to deny Doughty.
Despite throwing caution to the wind in the closing stages, Rob Edwards's side were unable to find the elusive third goal, ensuring they fell to their 12th league defeat season.
While the Hatters are guaranteed to end 2023 in the Premier League drop zone, Chelsea move up one place to 10th position.