A chaotic first half of football saw an early mix-up at the back for Chelsea, with former Brighton goalkeeper Robert Sanchez coming off his line to claim a looping ball but being beaten to it by Georginio Rutter, who headed into an empty net.
At the other end, Palmer was involved in everything for the Blues, hitting the post and seeing a would-be goal flagged offside before racing in behind to bury his first Stamford Bridge goal of the season after an Adam Webster error.
Brighton’s high line was presenting the hosts with so many chances, and after Jadon Sancho was denied his first Chelsea goal by another offside flag, the Blues took the lead from the spot.
Sancho played a pivotal role, drawing a foul from Carlos Baleba, allowing Palmer to step up to net his 10th Premier League penalty for Chelsea.
Just 10 minutes after netting his first, Palmer completed his hat-trick with a magnificent long-range free-kick.
However, a breathless first half continued as Brighton capitalised on another Sanchez mistake, cutting the deficit when Baleba intercepted a loose pass and tucked inside the near post.
The Chelsea shot-stopper wasn’t without merit, twice keeping out Baleba before yet more drama at the other end almost saw Noni Madueke add a fourth for the home side.
Fortunately for Enzo Maresca, the Blues soon restored their two-goal lead as another defensive lapse led to Palmer converting.
Despite only claiming 36% possession in the first period, Chelsea continued in the ascendancy following the restart, knocking on the Brighton door as Nicolas Jackson rounded Bart Verbruggen only to see his effort cleared off the line by Webster.
The hosts were peppering the Seagulls’ goal, and Palmer could easily have added a fifth goal inside the hour, seeing a dipping volley sail over the bar and placing another effort marginally wide.
A relatively calm period of play followed before Jackson again narrowly missed the chance to get his name on the scoresheet.
Former Brighton defender Marc Cucurella thought he’d netted from a corner in his last contribution of the afternoon, but he was also denied by an offside flag.
The second half was becoming all about Jackson, with the striker unable to convert one of his many chances, but thankfully for Maresca’s side, it proved inconsequential as they saw out a notably quieter second period.
Three points lifts Chelsea up to third in the early PL standings, just one point off behind leaders Manchester City.
As for Brighton, their unbeaten start comes to an end to leave them eighth, having won just two of their last 16 PL away matches.
Flashscore Man of the Match: Cole Palmer (Chelsea)