The Black Cats looked doomed just a few weeks ago when they slipped seven points adrift of 17th place in the Barclays Premier League, and few observers gave them any chance of avoiding the drop.
However, a return of 10 points from a possible 12 - from trips to Manchester City, Chelsea and Manchester United either side of a must-win home clash with Cardiff - has rekindled their hopes to such an extent that even a point against West Brom at the Stadium of Light on Wednesday night would almost certainly be enough to preserve their top-flight status.
But having seen Crystal Palace launch an astonishing late fightback to deny title-contenders Liverpool victory on Monday evening, Poyet will leave nothing to chance.
Asked if he ever feared Sunderland had passed the point of no return, he said: "We are realistic - I have said that word here many, many times, 'realistic'
It was becoming more and more difficult.
"When you have got 10 games, you think, 'If we win four or five, we are going to be all right'
Then you have got six and you are still there - it's more difficult, no doubt.
"We hadn't won three games in a row in the whole season and we have done it now in style - Chelsea, at Old Trafford and at home against Cardiff when that was probably the biggest game of all.
"That's football, I suppose
Listen, after the game yesterday, I think we all agree that anything can happen at any time, so you need to be spot-on, especially in the Barclays Premier League.
"You need to be at an incredible level because even in other parts sometimes when a team goes 3-0 down, they think 'Let's lose 3-0 now, there's no chance we are going to get back'
But here, yes, you can come back.
"It was good that game for us, it was good
It was a little bit back to tension, back to tomorrow, to the end, to the last minute like we did at Old Trafford.
"One-nil or 2-0 or whatever score it is if you are lucky enough to be up, you need to really work hard to keep that result because it's a great opportunity."
Should Poyet complete the mission he was handed in October last year, he will head off on his summer break to enjoy a well-earned rest, although he admits there will still be hard work to be done before he can do that.
He said: "There are so many things to do, I don't know where I am going to start
But depending on Wednesday or Sunday, of course the first thing is the players.
"There are plenty of players who have been in a tricky situation without contracts, and they have been playing and they have been fighting and they have been suffering and playing with pain, and I think they deserve my first spare time, so that's going to be the first move.
"Then, I suppose, pre-season, and then a little bit of relaxation with the wife - she's going to kill me as well."
Source: PA