The Seagulls were denied a replay when Theo Walcott's late goal gave the Gunners a 3-2 win after Brighton had come from behind twice.
"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 (Jack) 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."