Leonardo Ulloa scored the winner three minutes from time after El-Hadji Diouf had equalised Will Buckley's opener, only to be immediately sent off after gesturing towards the Albion fans.
Leeds had already seen Rodolph Austin red carded earlier in the game, while Brighton substitute Inigo Calderon was also dismissed in a highly-charged encounter at Elland Road.
"I am very pleased with the result. The Championship is becoming more and more difficult so we are delighted to be in the play-offs," said Poyet.
"We've lost Bruno Saltor injured so we are looking for a right-back to play in the next game and there are plenty of volunteers."
Calderon was sent off for fouling Diouf to give away the penalty which he duly scored from but Poyet was philosophical about the incident.
"Inigo will miss one match. If the penalty award was correct then probably the referee had to send him off, so it is about analysing the build-up to the penalty," he said.