The England international became the first player in Premier League history to score four goals in the first half of a match, helping Chelsea maintain their strong start to the season under Enzo Maresca.
Brighton initially took the lead just seven minutes in, capitalising on a chaotic moment in the Chelsea defence.
Georginio Rutter scored his first goal for the Seagulls after Robert Sanchez failed to deal with an aerial ball, with Moises Caicedo and Levi Colwill unable to clear the danger.
However, Chelsea responded quickly, and it was Cole Palmer who took charge.
He equalised by sweeping in Nicolas Jackson’s pass before doubling his tally with a calm penalty after Jadon Sancho was fouled in the box.
Palmer then completed his hat-trick in spectacular fashion, bending a sublime free-kick from 25 yards into the top corner.
Despite Sanchez’s second error of the day, which allowed Carlos Baleba to reduce Brighton’s deficit, Palmer wasn’t finished.
He restored Chelsea’s two-goal lead, drilling in his fourth goal after another assist from Sancho. Palmer could have added to his tally before the break, hitting ... and having a goal ruled out for offside.
The second half was less eventful, but Chelsea had already done enough to secure three points.
Brighton’s unbeaten run ended as they dropped to eighth place, while Chelsea moved to third, just one point behind Manchester City and Arsenal after six games.
Palmer’s brilliance left a lasting impact on Stamford Bridge, as the Blues continue to thrive under their new manager, Enzo Maresca.