The on-loan Liverpool striker was involved in two of his side's best attacking moments during the first half but did not emerge for the second half as Steven Fletcher came on in his place.
It later emerged Borini had gone to hospital but Poyet said: "He is feeling better.
"We have more information, the doctor called me. He is not going to fly back with us, he will stay down here overnight, but he has been released from hospital. Luckily it is nothing serious.
"We were really, really worried as he was really, really sick so the doctor decided rightly to take him to hospital but he has been released now. It was a virus or sickness so no problems."
When asked if Borini had collapsed, Poyet said: "He was a little bit sick before the game, not even this afternoon, but he wanted to play.
"Sometimes you have decisions to make between a manager and a player and we looked at each other and he said he wanted to play.
"He did well, he had a couple of opportunities, but at half-time he looked down and weak. We changed him, we did not give him a choice, then the doctor checked him twice and decided to have an extra check in hospital."