The 45-year-old praised his players following their 1-0 win over Manchester City on Sunday and says his team must believe they can avoid the drop this term
Phil Bardsley struck his second goal of the season in the 21st minute to sink Manuel Pellegrini's men and put Sunderland three points from Stoke City and West Ham, who are only out of the relegation zone on goal difference.
The triumph was Poyet's second win in four Premier League games since taking charge at the Stadium of Light in October and the 45-year-old says his side now have the belief that they can beat the drop.
"They deserve all the credit," he said. "They are the ones on the pitch. We can plan; we can talk; we can train but at the end of the day the players are on the pitch and need to do it and they were outstanding.
"Tactically it's difficult to explain but we talked a lot, and tried to convince them it was possible - especially to keep a clean sheet against a top, top team who have scored seven and five in their last two games.
"It just shows that if we believe and we are convinced, I think it [avoiding relegation] is going to be possible."
The Uruguayan reserved special praise for Wes Brown after the defender produced a solid display in his first league start for 22 months.
Poyet added: "Credit to him [Brown] - I was thinking to put him in straight away when I arrived at the club, but for someone reason we felt he was two or three weeks away. "