The Black Cats return to league action against fellow strugglers Crystal Palace on Saturday sitting inside the bottom three with 12 games remaining and knowing the survival fight in which they have been engaged almost from the start of the season is likely to continue until its death throes.
Poyet inherited a side which had claimed just a single point from the first 21 it had contested by October and has managed to boost the total to a more respectable, but still insufficient 24 over the 19 league fixtures to date of which he has taken charge.
That, coupled with runs to the Capital One Cup final and the last eight in the FA Cup, has increased his conviction that salvation remains in their own hands.
Poyet said: "That's why I am here. If not, I would be gone, I can promise you that. I wouldn't stay here if I was not convinced it was possible.
"I am not that type of person that would just lie to myself and lie to people, 'Yes, yes, yes, it's possible'.
"Maybe I am making a mistake, who knows? But again, I believe it's possible because these 19 games I have been in charge have proved to me - we have taken 23 points - it's possible.
"If we take 13, 14 points [from the remaining 12 games], it's going to be close to that 40, 37, 38, 39 - who knows?"
Poyet has set his players a target of five victories to ensure their continued presence in the top flight, although the fact that they have managed only six so far is a measure of the problem that could present.
There have, of course, been high points during a difficult campaign with a derby double over Newcastle, a league victory over Manchester City and cup successes against Chelsea and Manchester United providing evidence that the Uruguayan's methods are working.
However, there have been bad days too, and the 46-year-old knows his team cannot afford too many more of those.
He said: "If you said to me on October 8 when I walked in here at 11.45pm for the first time that I was going to be in this situation with 12 games to go, I would have taken it, I would have.
"I would have for sure because it was looking impossible, so now I am here, I am not going to complain and moan.
"I am taking it and making sure we maintain the level of performance that we have had in the last three months and if we maintain that, I think we have got a great chance to stay up."
Whatever happens over the next two months or so, there will be work to be done during the summer, and much of it will come under the remit of newly-appointed sporting director Lee Congerton.
Poyet said: "I am convinced the most important part in football nowadays is recruitment.
"If you recruit well, a big, big part of the manager's job is much easier; when you recruit averagely, you have got a massive job and if you recruit badly, all the best.
"That relationship between manager or head coach and sporting director or head of football operations, whatever you want to call him, is a key part, massive.
"It needs to be clear and honest and straightforward. It's not like one is working for the other.
"It's simple: He needs to bring the best players for me to be able to play the football that I want and to be successful."