Poyet was unhappy at Howard Webb's refusal to point to the spot when Kevin Nolan elbowed a loose ball in the penalty area at the Stadium of Light.
Sunderland were still trailing at that point, after Andy Carroll's early header, before the hosts' woes were compounded when Mohamed Diame's deflected strike went in five minutes into the second half.
Substitute Adam Johnson – bizarrely left on the bench despite being Sunderland's top scorer – produced a response with an excellent 20-yard effort in the 65th minute.
But it was too little too late as Sunderland remained 19th, four points adrift of safety, as their winless run extended to seven matches in all competitions.
Asked if he thought it was a penalty, Poyet told Sky Sports: "Yeah. Yeah, it was clear. It was deliberate.
"He (Nolan) tried to play the ball with his hand. I think Howard was in a good position… perfect angle, easy to see.
"I think it is one of them where you need to ask the referee."
Sunderland face Tottenham, Everton, Manchester City and Chelsea – who are all in the top six – in their next four Premier League matches.
But Poyet has vowed his side will keep fighting.
"It's difficult to accept (these defeats) because sometimes when you see a team down at the bottom, a team that is suffering and with no spirit – they just give up," he said.
"I don't think it is possible to see one of my teams giving up so it is going to the end, that's for sure.
"I think today we showed what we are, and that's it. Sometimes you need to be realistic and know the team we have got and know what you can do.
"I think today we tried everything possible. I hope we can get the luck… and we can win a few more games. It's about winning now.
"We still have a chance and we are going to go to the end."
