Glenn Murray scored a controversial penalty to put Brighton ahead and the host added another two after Shane Duffy was dismissed for pushing his head into Patrick van Aanholt's in the 28th minute.
The red card prompted the introduction of Leon Balogun and he netted with his first touch, while Florin Andone, who came on after Murray sustained a shoulder injury, struck for the second match in succession just before the break.
Luka Milivojevic converted from the spot nine minutes from time, but Palace was unable to avoid slipping to a disheartening defeat after a promising draw at Manchester United and win at Burnley in its previous outings.
The visitor was unfortunate to fall behind in the 24th minute when referee Kevin Friend deemed that James McArthur had fouled Jose Izquierdo despite winning the ball and Murray guided the resultant penalty home.
Friend did not award a spot-kick when James Tomkins sent Murray to ground and while the striker received treatment on his shoulder Duffy was shown a red card for a clash with Van Aanholt.
Chris Hughton sent Balogun on in response and the Nigeria international lashed a brilliant right-footed strike past Wayne Hennessey just 25 seconds later to complete a remarkable string of events.
Murray had to be replaced by Andone and, after a brilliant goal-line clearance from Bernardo kept Milivojevic out, the Romanian punished Palace on the counter-attack in the fourth minute of stoppage time.
Brighton tried to hold out for a fourth Premier League clean sheet of the season but was unable to do so as Milivojevic scored a penalty for Palace after Wilfried Zaha was fouled by Balogun, though it was nothing but a consolation.