The Seagulls were drubbed 6-1 at Liverpool in the fifth round last term but gave Arsenal a fright as goals from Ashley Barnes and new signing Leonardo Ulloa cancelled out Olivier Giroud's double.
But Theo Walcott grabbed a late, deflected winner for the Gunners, leaving Poyet wondering just how to deal with the England forward.
"In some parts I am pleased because we competed better than last year at Liverpool. Much better," said Poyet.
"On the other hand we had a great chance to make some news, but it wasn't to be.
"I saw one sprint from Theo Walcott and it was scary. How to stop him? Maybe shoot him?
"You play 70-75 minutes and even with Wayne Bridge, you have to compete with someone sprinting like that in the last 20 minutes, and Jack Wilshere going past people in the middle. But the biggest problem was Giroud.
"Possibly we should have been ahead at half-time but that's football. I knew we would concede one or two, I didn't expect three. We conceded a late deflected goal. It wasn't to be."