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.
Despite having only 10 men from there, the hosts responded strongly and Mannone produced a series of impressive saves in the second half to help keep Everton at bay.
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."
The build-up to the penalty being awarded saw Howard opt to play a short goal-kick to Leon Osman, who miscontrolled the ball.
That allowed Ki to claim possession and dart into the area, where he fell to the turf when Howard came out to tackle him.
Regarding that incident, Poyet said: "Everything changed with the mistake.
"I always say that sometimes I like games without accidents, but that was a big accident for us to take advantage of.
"Then, probably because of the situation we are in and probably because of how good Everton are, even with them having 10 men, we were just defending for our lives.
"This is the best team we have played against, even with 10 men."
Fifth-placed Everton went into the contest unbeaten in 10 matches and having not lost a league game at home in the whole of 2013.
Poyet added of the surprise Boxing Day triumph: "This is the best present you can ever give me, three points.
"It is a good day. It was extremely difficult against 10 men, but football is incredible and you never know what is going to happen."