The Seagulls flew out of the blocks and were ahead inside 80 seconds when Ashley Barnes tapped home - Andrea Orlandi's header doubling the lead soon after to render Jeff Hendrick's reply mere consolation.
Poyet, whose side have now won three in a row, said: "To come back with three wins is down to the players and they've been a bit less pretty.
"We took advantage of the situation in our FA Cup game against Newcastle and need to keep going. It's up to me to manage the squad well and keep us strong.
"There were some nerves and it was our fault really. But it was three points at home and that was the key. If you don't win, you're not good enough.
"If you are organised and make it difficult for the opposition, that's great. We have got a good squad and it's there for us. It's been a good start to the year."