The Black Cats secured a vital 1-0 win at high-flying Everton thanks to Ki Sung-Yueng's penalty.
The spot-kick came after Everton goalkeeper Tim Howard brought down Ki following a mistake by Leon Osman. The midfielder had miscontrolled the ball just outside the area and the Korean nipped in to steal the ball before taking it around Howard, who was subsequently sent-off.
Despite the victory, Sunderland, who are now unbeaten in four games, still sit bottom of the table after fellow strugglers Fulham and Crystal Palace also won.
But Poyet believes his side can still get out of trouble, especially if they can pick up a win in two massive games in the coming week against fellow strugglers Cardiff and Aston Villa.
He told Sky Sports: "Mistakes, sometimes they define moments in the game and it was a big mistake for them.
"We took advantage and I'm pleased that we were able to hold today because you saw the reaction of the fans. It means so much for them.
"We didn't win too many this year, so we were trying to finish 2013 - it didn't look a good year for us - but it looks like we're going to finish a bit better."
He added: "We are still down in there, at the bottom, but I'm sure if we can pick up another win between now and the 1st of January, the second part of the season is going to be very good."