Goalkeeping errors at both ends, high defensive lines, and speedy counter-attacks made for an exciting match, with Brighton's French forward Georginio Rutter opening the scoring in the seventh minute. However, it was 22-year-old Palmer who stole the show with an impressive hat-trick in just 11 minutes.
Howls of derision from the Brighton supporters greeted their former goalkeeper Roberto Sanchez who came out of his goal in a scramble and missed the ball, giving Rutter a free header.
But Chelsea, now the league’s top scorers this season, turned on the pressure and Palmer hit the post and had a goal disallowed for offside before his scoring spree.
The first after 20 minutes came from a Nicolas Jackson interception. The Senegal striker raced on goal before squaring the ball to Palmer who tapped home.
Palmer’s second came from the spot when he sent Bart Verbruggen the wrong way after Jadon Sancho was brought down in the penalty area.
The former Manchester City trainee has been dubbed “ice Cole” and he showed his cool once again with his 10th successful penalty in 10 attempts since joining Chelsea last season.
Palmer completed his hat-trick on the half hour with an elegant free kick from some 30 metres out which curled perfectly beyond Verbruggen into the top corner.
Carlos Baleba pulled a goal back for Brighton in the 32nd minute after a second error from Sanchez, who fluffed a clearance.