The Black Cats eased their relegation worries with a 1-0 win over Hull last Saturday and can take a giant step to safety with victory at bottom side West Brom this weekend.
This current relegation scrap comes after the Wearsiders earned a hard-fought 15th-place finish under Roy Keane last term.
Leadbitter, a local lad, says the team want to push on for new boss Ricky Sbragia and bring success to the club.
He said: "We all want to stay in this league. The Premier League is the best league in the world and everyone wants to play in it.
"Everyone wants to stay in the league, not just for ourselves and the club, but also for Ricky.
Man-manager"He is a man-manager and he makes you want to be the best player that you can be. He is popular in the dressing room and we all want him to succeed.
"We want to stay in the Premier League and then push on next season and make sure we are not fighting relegation again."
Leadbitter is targeting victory at The Hawthorns to add to the vital win against the Tigers.
He said: "We needed our big-name players to step up against Hull and they did that.
"It was a vital three points - but it will mean nothing if we don't go to West Brom on Saturday and get something there.
"One result in isolation isn't enough, not at this stage of the season.
Run"We have to follow it up and, hopefully, get a four or five-game run going where we pick up at least a point per game.
"If we can do that, we will get ourselves safe."
While the Baggies are currently rooted to the bottom, Leadbitter is not underestimating the threat posed by Tony Mowbray's side.
He told the Sunderland Echo: "Everybody thinks West Brom are as good as down, but they are not yet. They are still fighting for their lives.
"You never know what will happen in football - if they can put three or four wins together, then they have to believe they might get out of it.
"It won't be easy on Saturday. West Brom are a bit unfortunate to be in the situation that they are. I think, of the sides that came up, they are possibly the best footballing side of the three.
"They have shown all season they are capable of scoring goals, but they concede them, too, and that is what we have to capitalise on."