Jason Scotland and Chris Dagnall scored either side of Leonardo Ulloa's leveller, before the Brighton striker saw his late penalty saved.
Brighton have taken just one point from the last nine available but Poyet refused to point the finger elsewhere.
"I was convinced we had a great chance and I believe we lost the chance of
getting three points," Poyet said.
"We knew we would have to fight and compete and I got the feeling that we gave the game away with the goals that we gave away, but that is how it has been for a while so I am not surprised.
"Unfortunately I am getting used to this, we have played three games away from home in a week. The first game was terrible, then we were by far the better team at Bolton and lost 1-0 and today if anyone said Barnsley deserved to win then I should retire.
"The good thing is the only people to blame is ourselves. It's not the opposition, the pitch, the referee or the weather. We have another nine games and we'll see. We'll try our best."