Poyet though was pleased with his team's performance overall and believed they were unlucky not to come away with a point after a grandstand finish in which they came so close to grabbing an equaliser.
"Today my players gave a great lesson on how to defend for so long, wait for your chances and we nearly made history," the Uruguayan said after the game. "We were just a bit short in the final finish, but that is football. If we had scored it would've been the perfect way to play with nine players against 11.
"My team understood what they had to do. They listened to what I said at half-time and it was a really brave effort, and a class act from all the players. The way they marked, blocked shots and did everything they needed to do made me proud.
"We almost went up the other end and had a spectacular finish but it wasn't to be. Sometimes teams do nothing for most of the game and still win, but we did everything we could've tonight and I'm very proud.
"The fans were also magnificent as they really tried to help the players. I just wish we could have sent them home with the point we deserved. I'd like to wish all fans a very Merry Christmas with their families, and I hope they can forget about referees."
He opted not to discuss the performance of referee Craig Pawson and added: "I'm not going to comment on referees. If I say anything I might get in trouble and I don't like wasting money, therefore I'm not going to comment.
"I have sympathy for the fans who paid good money to come into the ground, but like us, they have to be brave and move on. All the reporters can say something, they have been here today and seen what happened, but I'm not going to give them anything to talk about."
Having seen the Clarets pick up a third successive away win, Eddie Howe admitted: "I've never been in a game like that before. It was the strangest atmosphere to a game that I've known and the lads on the pitch felt it as well.
"The Brighton fans were excellent and stuck behind their team and all their frustration was vented towards the officials, which, if they are sendings off is harsh on them. The fourth official was in a clear view (for the first) and he saw it.
" I didn't actually see it, but he had the same view that Jason (Tindall) had and saw it as a clear sending off. But for me to comment on either sending off though, without looking at the incidents again and reviewing them would be foolish.
"Then we got the lead and didn't have to chase so much any more but we always knew there would be a period in the game though when Brighton would go for it and we wanted to get that second goal.
"There was a little bit of fun at the end when Brighton came at us, but we managed to hold on. We are just pleased to have got the win and the three points."
He added: "The pleasing things from my point of view is that the lads kept their heads. It would have been easy to get drawn in and the crowd would have liked the referee to even things up.
"The main message at half time was to keep our heads and keep our discipline to retain our numerical advantage. The easiest thing in the world would have been to get caught up in it. Brighton felt aggrieved and were trying to get reactions from our players and I have to say, we kept our nerve and discipline well for a young side, which is pleasing.
"All in all it was a really strange game and a difficult one for us. People will say we had two men to the good and it should have been quite comfortable, but Brighton, to their credit, defended very, very well and made it difficult.
"They isolated Mackail-Smith from his team mates and defended the 18-yard box and it was like a training session of attack v defence. I thought we probed and we passed and created chances, but it was really difficult because all the pressure was on us and we could only mess it up.
"That's four out of five wins now, which is a great statistic and now we go into these two home games looking to keep going in the right direction."