The Black Cats head for Tottenham on Monday night lying seven points adrift of safety and knowing they will need to win at least three of their remaining eight games if they are to stand any chance of remaining a Barclays Premier League club.
Given the fact that they have only six league wins to their name to date this season, and that their four away fixtures take them to Spurs, Manchester City, Chelsea and Manchester United, that may be a tall order.
However, Poyet knows from experience that strange things happen as seasons draw to a close, and he is hoping that will be the case for his struggling team.
Asked if they may have to do something unexpected to survive, Poyet said: "Yes, yes - but I think it is possible.
"I remember clubs in the past - I am not going to talk about names - they have played five or six games at the end of the season against teams who were playing for nothing and they won four or five.
"People say it's incredible, a miracle, but it depends what the there teams put on the pitch and what you do. But it can happen.
"I suppose I would try as a professional not to alter the end of the season, but if a team goes down early and they make seven changes to try things for the following year, I wouldn't blame them because they are trying for themselves.
"Now that can affect the result of that game and then everybody is blaming somebody else. But a the end of the day, you have got 38 games, it's you.
"You cannot blame the rest for what they do because it happens during the season when someone is playing in the Champions League and chooses to make five changes in midweek.
"Nobody remembers those, they only remember the last three or four when people are already out of any possibility."
Possibilities and time, however, are running out fast for Sunderland, who have won none of their last six league games, a run which culminated in a damaging 2-1 home defeat by West Ham on Monday evening.
Winger Adam Johnson, whose appearance as a second-half substitute in that game inspired a fightback which came up only just short, admitted afterwards that the point of no return had been reached, and Poyet agrees.
He said: "No time ago, we had 12 games to go, and then it was 10 and now it's eight and we haven't won...
"Yes, I agree with him. Now if you say it clear and loud, it's pressure for the players - but if we are not under pressure by now in terms of winning the next game, I don't know when we are going to be.
"Of course yes, I totally agree. It's good that the players say it as well because they are inside here every day and they know what can happen."
Poyet, of course, inherited a difficult situation from predecessor Paolo Di Canio with the Black Cats having taken just a single point from their opening seven league fixtures.
As a result, he has had to cope with intense pressure throughout his reign to date.
He said: "Me, I am realistic, more realistic than anyone. I don't forget when I got here. I didn't write it anywhere.
"I made a mistake not putting it on my fridge so every morning when I open the fridge, it reminds me that we had one point from seven games and we were there.
"I didn't, but I made a mistake there. But I remember, don't worry."