The Black Cats have just one point from their first seven games and currently sit rock bottom, six points from safety.
"I have no doubt [we can survive]," Sky Sports News quotes Poyet as saying. "The idea is to believe. The key is to convince the players. They are the ones who need to feel on the pitch that they have got a chance to win the game.
"It's well-known in football that when you are at the bottom, you get no luck, no help, everything goes against you. We need to expect that, we need to be ready for that.
"Whatever happens, it doesn't matter. We need to keep believing that this is the way. I am responsible for that. The key is that they are convinced and they believe that is possible. We will see."
Sunderland travel to the Liberty Stadium to take on Swansea City on Saturday.