Player-boss Andy Morrell's stunning first-half strike threatened another famous Red Dragons giantkilling, 20 years after their win over Arsenal, but Ashley Barnes headed an equaliser with 13 minutes remaining to level the score at 1-1.
Dean Keates then missed Wrexham's first penalty and Brighton converted all their spot-kicks to triumph 5-4 and secure a fourth-round visit from Newcastle.
Poyet said: "It was a very difficult game. We knew we did not start well but the cup is about getting through somehow.
"We did not adapt well to the conditions in the first half, then we started to play a little bit better.
"We created the chances and scored at the right time and then it was very open. I don't like open games. But that is the cup. Then on the penalties the experience and quality helps a lot."