Kieran Trippier's goal was overshadowed by red cards for Romain Vincelot and Ashley Barnes, as well as the touchline antics of manager Gus Poyet.
Vincelot was first to go on Saturday after becoming involved in an off-the-ball altercation with Marvin Bartley just six minutes in.
The incident was brought to referee Craig Pawson's attention by the fourth official but many in the ground, Vincelot included, seemed mystified by the decision.
It got worse for the Seagulls six minutes later when Barnes stamped on Chris McCann, prompting an angry reaction from the midfielder, and was handed a straight red.
The dismissals sandwiched one of the few shots at goal in the first half, Kazenga LuaLua narrowly clearing the bar from long range.
Brighton boss Poyet made two changes midway through the opening half, bringing on Craig Noone and Alan Navarro for LuaLua and Ryan Harley.
Despite their numerical disadvantage, Brighton tested the visitors, with Noone's penalty-box tumble giving the referee another decision to make, but he waved away appeals for a spot-kick.
It all got too much for Poyet, who threw his coat to the ground and stormed down the tunnel. His mood would have worsened drastically when Trippier opened the scoring after 32 minutes.
The defender stepped inside and fired in a powerful effort from the edge of the box for his first league goal.
With the hosts understandably sitting deep, Burnley were afforded a lot of possession but they were almost stung when a quickly-taken free-kick set Craig Mackail-Smith away just before the break, however Michael Duff got back to clear.
Having won their last three home league games, Brighton showed resilience in the second half, attempting to put the Clarets under pressure from set-pieces while working hard off the ball to restrict their opponents.
It was still Eddie Howe's men who created the better chances, however, with Ben Mee's 35-yard strike producing a good save from Casper Ankergren.
With 15 minutes left, Ross Wallace tried his luck from long distance but Alan Navarro was in bravely to block.
Burnley did not have it all their own way and, spurred on by a boisterous home crowd, Brighton went close to an equaliser with two efforts inside three minutes. First Mauricio Taricco tested Lee Grant with a long-range effort before Mackail-Smith drilled a shot narrowly wide from inside the box.
Suddenly, Burnley looked happy to preserve their lead when earlier their intentions seemed more positive.
That approach invited pressure and it took a goalline clearance from David Edgar to deny Mackail-Smith in injury time.
Burnley rode out the remainder of the match with a nervy performance to record their fourth win in five games.