Chelsea, who had Reece James sent off late on, have now won their last four league matches during an excellent run of form, with the result against Roberto De Zerbi's side moving them onto 60 points from their 37 matches.
Danny Welbeck netted in the final stages for Brighton to potentially set up a grandstand finish, but the visitors navigated their way through the final stages to secure a huge three points.
The Blues have moved above Newcastle United into sixth position in the table, and they are now three points ahead of the Magpies and just three points from fifth-placed Tottenham Hotspur, with Mauricio Pochettino's team still capable of finishing ahead of the North London outfit.
Chelsea have all but secured a top-seven finish due to their vastly superior goal difference over eighth-placed Manchester United, and the London club will be bidding to continue their excellent run of form and secure a European spot for next term when they host Bournemouth on Sunday afternoon.
Brighton, meanwhile, will be bidding to bounce back when they welcome Man United to the Amex on the final matchday of the 2023-24 Premier League campaign.
Chelsea can still finish fifth in Premier League
Chelsea were bright early on, with Brighton goalkeeper Bart Verbruggen needing to be alert to keep out an effort from Noni Madueke, who had driven into the home side's penalty box.
The hosts were also threatening, with Simon Adingra's effort from a Tariq Lamptey cross leaving Chelsea goalkeeper Djordje Petrovic rooted to the spot, but it just dropped wide of the post.
Verbruggen also denied Chelsea forward Palmer in the opening 15 minutes, with both sides having their moments at the Amex.
Chelsea were awarded a penalty in the 16th minute when Facundo Buonanotte challenged Marc Cucurella inside the penalty box, but referee Michael Salisbury changed his decision after checking the pitch-side monitor, with VAR preventing what would have been a controversial call.
Verbruggen did well to deny Malo Gusto in the period that followed, before Adingra just missed the target with a curler, and it was not a surprise that the opening goal came during a lively period of the contest.
Palmer made the breakthrough, with the Englishman registering his 22nd Premier League goal of the campaign in the 34th minute by brilliantly heading a Cucurella cross into the back of the net.
Palmer continues excellent scoring form
The 22-year-old is now on 25 goals in all competitions for Chelsea since his arrival from Manchester City, and he has to go down as one of the best signings of the season.
Chelsea had the ball in the back of the Brighton net for a second time in first-half stoppage time, but Nicolas Jackson's header was ruled out for a foul on Lamptey, before the Blues forward volleyed wide of the post from close range in the final stages of the opening 45 minutes.
Petrovic also needed to tip a Joao Pedro header over the crossbar before the half-time whistle arrived.
Brighton should have levelled the scores early in the second half through Pascal Gross, but the midfielder somehow missed the target from close range.
Instead, Chelsea doubled their lead in the 64th minute through Nkunku, who carefully converted brilliant cross from Gusto, and Brighton were unable to launch a response.
Pochettino's side have come from nowhere to now put themselves in pole position to claim a top-six spot, and there are plenty of reasons for the club's fans to look forward to a brighter future.
Wednesday's contest ended on a negative note for Chelsea, though, with James given a straight red card for a kick on Pedro, with the referee sent across to the monitor again after advice from VAR, before Welbeck netted a late consolation for the home side, tapping a low Pedro cross home from close range.