Theo Walcott hit the winner five minutes from time after stepping off the bench, with Olivier Giroud's earlier efforts cancelled out by Ashley Barnes and Jose Ulloa.
The Seagulls also had to cope with losing both captain Gordon Greer and winger Will Buckley, who had stretched Arsenal down the right, to injury.
"It is a funny feeling because we are more disappointed than we should be in the dressing room. It is like we had lost a final," Poyet said.
"We had the chance to do something special, but when you lose players like we did to injury, against players like the quality of Arsenal, there is not much you can do.
"When you see Wilshere doing stretches and Walcott sprinting up and down on the sidelines ready to come on, you know you have got a problem.
"It was difficult, but we did what we had planned - competed and scored two good goals."