The Black Cats were able to take only a point from a 0-0 draw with Crystal Palace at the Stadium of Light on Saturday, a game which Poyet had admitted was a must-win affair, and are now three points adrift of safety, although with two games in hand on their closest rivals.
However, the Uruguayan insisted there was still too far to go to for anything to be decided just yet.
He said: "Nothing will change until the last game, nothing. Apart from some big surprises, I don't think a team is going to been gone in the next four or five weeks, completely down or the opposite, completely safe.
"It's going to be tight. It could be a goal, which sometimes people don't think. It could be one goal, the difference between staying up in the Barclays Premier League or going down, so we need to be smart.
"It's not going to be easy, but if we try like today and we fight like today, we have got a better chance, that's for sure."
Sunderland head for Norwich next weekend having passed up what they believed was the perfect opportunity to secure just a second home league win in eight attempts.
They failed to make the most of a series of half-chances which came their way before the break, three of them falling to striker Steven Fletcher, but went close to the opening goal within six minutes of the restart.
Half-time substitute Jozy Altidore initially miscontrolled Wes Brown's long ball, but having regained possession, span and fired in a shot which Palace goalkeeper Julian Speroni had to tip over.
However, it was Fabio Borini who came closest to breaking he deadlock as time ran down, rattling the crossbar with a stinging 83rd-minute shot and then only just missing the target when the ball fell to him once again.
Kagisho Dikgacoi might even have won it for the visitors at the death, but blasted his effort inches wide to spare the Black Cats further punishment.
Asked for his reaction, Poyet, who revealed Fletcher had been withdrawn because of a twisted ankle which will be assessed over the next 48 hours, said: "A word you will probably never hear again from me, very calm. I feel calm today.
"We tried everything. That's what we are, that's the team that we have, the players we have got and the way that we need to play.
"If we play like that, the future will tell if we are good enough to stay in the Barclays Premier League or not.
"But that's the way to perform, to take the game to the opposition, to assume risk, to be in control, to try crosses, free-kicks, shots, in behind, in front, one-twos - we tried everything."
Palace boss Tony Pulis was understandably the happier of the two managers, but admitted the Eagles too remain firmly entrenched in a battle for their top-flight lives.
He said: "We are 17th, that's where it leaves us, and it leaves us in a situation where we have to try to get as many points as we possibly can.
"We are in the frame with 10 other teams - it's remarkable, really, how many people are down there and how many clubs are involved in it.
"It was a must-not-lose game for us, very much so. The last two away games, it has been two clean sheets and it's been what we have worked on."