Dragons player-manager Andy Morrell's stunning first-half strike threatened another famous Wrexham giant-killing, 20 years after their win over Arsenal, but Ashley Barnes headed an equaliser with 13 minutes remaining to level the score at 1-1.
The non-league side missed several chances, notably when Jay Harris headed against the bar with extra time almost up, but ultimately they were made to pay for Dean Keates' penalty miss.
The Seagulls will now face Premier League Newcastle in the fourth round but Poyet conceded it was a barely deserved victory.
"It was a very difficult game," he said. "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.
"When you come here you have nothing to win and everything to lose. If you win 3-0 that's normal. If you win on penalties you are lucky.
"So it is not easy. Saying that, most of the time it is credit to the opposing team and you can see how much it means for the players.
"It was a great FA cup tie, I can imagine how Wrexham feel. They were so close to having a great game here, but that's football, sometimes it goes for you and sometimes it goes against."
The 61-place difference between Brighton and the Blue Square Bet Premier leaders was hardly visible and Morrell is hoping to use their performance as a springboard to promotion.
He said: "The lads were absolutely fantastic today [Wednesday] and they deserve all the credit that they get and on another day we might have nicked it as well.
"It wasn't to be, but I am so proud. We had the better chances I think, but I am not going to sit here and complain.
"They are devastated down there and I have told them I don't see why they should be.
"We have just given a decent Championship team a right run for their money, I can't fault any of them.
"We have got bigger fish to fry with getting out of this league and if we can do that with a good cup run this should kick us on now, rather than hinder us.
"I think we have a great chance of going up if we can keep the lads fit and together.
"I think this is a springboard, Crawley did it last year and it's a very similar thing, apart from they were at Manchester United, so I don't see why we can't."