Patrick van Aanholt scored with a 25-meter free-kick and Lamine Kone added two more to give Sunderland an unsurpassable four points lead over both Newcastle and Norwich with just one game to play on Wednesday, reports Xinhua.
Sunderland's survival depended a lot on the appointment of coach Sam Allardyce after Dick Advocaat resigned last October following eight games without a win.
Allardyce, who has never seen a Premier League side relegated in 25 years, saw the Black Cats lose only twice in their past 13 games.
Their victory leaves them four points ahead of both Newcastle and Norwich with just one game to play.
Norwich were relegated to the Championship despite a 4-2 victory over Watford at Carrow Road.
Troy Deeney had given Watford the lead, but Norwich responded through Nathan Redmond's strike, Dieumerci Mbokani's tap-in and a Craig Cathcart own goal.
Odion Ighalo pulled a goal back, but Mbokani dinked in Norwich's fourth.
In the day's other match, Christian Benteke's late header cancelled out Eden Hazard's brilliant solo goal as Liverpool kept alive their slim hopes of a top-six league finish.
Chelsea winger Hazard fired a low shot into the corner after a dazzling run to record his fourth goal in four games.
But substitute Benteke earned Liverpool a point in added time when he pounced after an error from Asmir Begovic.
Liverpool remain eighth and are within three points of sixth-placed West Ham.