Sunderland appeared to be heading for a 2-0 defeat until they launched a late rally and goals from Steven Fletcher and Jack Colback salvaged a precious point.
The Black Cats still sit at the bottom of the Premier League, but they have picked up six points in the last four games and Poyet insists his side will move away from trouble.
"I do feel we will be in the Premier League next season because we are back there with the other teams," he said.
"When I took over there was a big gap to the teams in the middle of the table, now we are one win away from getting out of the relegation zone.
"That is a great feeling for the players and the fans. It is now up to us, we are five games unbeaten.
"When you are at the bottom people do not notice that, but we had been defending well until this game and we scored today.
"We are going in the right direction so let's hope we can maintain that consistency in the New Year and get another three points.
"We have character and desire and we are going to fight."