Sunderland's winner came in the 25th minute via a penalty from on-loan Swansea midfielder Ki Sung-yueng, who had gone down in the box under a challenge from Tim Howard that earned the Toffees stopper a straight red card.
Asked after the match what had made the difference for Sunderland, who are bottom of the Barclays Premier League and picked up all three points for the first time in seven fixtures, Poyet said: "Concentration, belief.
"Maybe when nobody expects you to do something special, the players get together.
"And our goalkeeper was outstanding.
"Sometimes strikers get all the credit, but sometimes you need a goalkeeper to win you a game and today Vito probably won the game for us."
For breaking stories and all the great banter like us on Facebook: facebook.com/tribalfootball