The Seagulls came close to earning a replay in their FA Cup fourth-round clash, before Theo Walcott grabbed an 85th-minute winner for the Gunners.
Despite the defeat, Poyet was upbeat about his side's display, but admitted that they found it difficult to deal with Arsenal's quality.
"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," said Poyet.
"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."
Brighton are currently chasing promotion in the Championship, lying in seventh place.