Kieran Trippier's goal was enough to earn Burnley victory after Brighton had two players sent off early on.
Romain Vincelot was first to go after an off-the-ball row with Marvin Bartley before Ashley Barnes was dismissed for stamping on Chris McCann.
Despite their numerical disadvantage, Brighton tested the Clarets several times before the visitors struck.
Trippier hit the decisive winner when he fired a powerful shot from the edge of the box for his first league goal.
Referee Craig Pawson was busy throughout the match and also ignored appeals for a Brighton penalty after Craig Noone went down in the box.
The decision infuriated Brighton manager Gus Poyet, who threw his coat to the ground in anger.
Burnley created the better chances, however, with Ben Mee's 35-yard strike producing a good save from Casper Ankergren and Ross Wallace was denied from long distance by Alan Navarro's brave block.
Brighton twice went close to an equaliser with Mauricio Taricco and Craig Mackail-Smith almost levelling.
Burnley sat back on their slender lead and were fortunate when David Edgar denied Mackail-Smith in injury time with a desperate goalline clearance.
Brighton manager Gus Poyet:
"I don't want to comment on the sending off incidents, it's up to the club whether they want to take it further but I'm not because I don't want to spend any money. Some people in games like this may lose their jobs. It was a really bad day at the office.
Burnley manager Eddie Howe:
"It's sometimes harder with a numerical advantage and it was a good experience for our lads, a really good test for us. It was a difficult last five minutes. We didn't want to sit too deep and our defensive line wasn't great at the end. I can't recall playing for such a long time against nine men, but the atmosphere the Brighton fans created was terrific and if you stand off for one second someone like Craig Mackail-Smith will punish you."