The Black Cats drew 2-2 with Manchester City in midweek before beating Chelsea 2-1 at Stamford Bridge and the ex-Tottenham midfielder feels they can pull off a great escape
Having appeared almost certain to suffer relegation, Poyet's side have been given fresh hope of survival courtesy of two outstanding performances at teams contending for the title.
|9/1||Sunderland are 9/1 to beat Cardiff in their next fixture|
Fabio Borini's late penalty proved enough to condemn Chelsea to a first home league defeat under Jose Mourinho.
With four games left to secure safety, Poyet admits he does not know exactly what to expect from his team after only recently stating that they would need a "miracle" to stay up.
"We're going to go to the end. We're going to fight for them (the Sunderland fans)," said the former Brighton and Hove Albion boss.
"With results like today (Saturday) you can expect anything from us now, one way or the other.
"The only thing I want is to keep this belief and performance until the end of the season."
Poyet acknowledged expectations will be far different when Sunderland, who remain bottom of the Premier League, face 18th-placed Cardiff City at the Stadium of Light next Sunday.
"We accept that we are the smallest team (against Manchester City and Chelsea), we know," he added.
"Now we play Cardiff next week - how are we going to convince everyone in England that we are the smallest team against Cardiff?
"You tell me because I've got no chance. Everybody will expect us to beat Cardiff next week.
"That means that maybe we are the better team. Are we?"