Sunderland centre-back Lamine Kone's first two goals for the club and a Patrick van Aanholt free-kick took Sam Allardyce's side four points clear of danger with one game remaining.
Norwich returned to the Championship a year after promotion despite a 4-2 win over Watford, while Newcastle, Sunderland's arch rivals, will play in the second tier for the first time since 2010.
"It's the attitude of the players who absorb the pressure and deliver a performance like that," Allardyce told Sky Sports in an on-pitch post-match interview.
"That's what's happened over the past 10 games -- only one defeat. It's pretty miraculous. To the lads and all the staff, thank you very much."
It is the fourth season in a row that Sunderland have pulled off a late escape to avoid relegation and it guarantees the northeast club top-flight football for a 10th consecutive season.
Allardyce was the architect of their surge to safety, succeeding the sacked Dick Advocaat in October and overseeing a late-season run that has seen them take 11 points from a possible 15.
Having seen his side come from 2-0 down to beat outgoing champions Chelsea 3-2 on their previous outing, Allardyce kept faith with the same starting XI for the seventh game running.
His side went ahead when Everton goalkeeper Joel Robles misjudged Van Aanholt's 38th-minute free-kick, moving to his left and allowing the ball to bounce straight past him.
Kone made it 2-0 four minutes later, volleying in emphatically after Sunderland failed to clear a corner, and smashed in his second in the 55th minute when Robles spilled Wahbi Khazri's corner.
The result at the Stadium of Light also served to increase the pressure on Everton manager Roberto Martinez, whose 12th-place side have won just one of their last 10 league matches.
- Newcastle 'devastated' -
Alex Neil's Norwich got the better of Watford in a ding-dong contest at Carrow Road, but Sunderland's win meant that they would have been relegated regardless.
After Troy Deeney had tapped in Odion Ighalo's 11th-minute cross to put Watford head, Norwich hit back through Nathan Redmond, Dieumerci Mbokani and a Craig Cathcart own goal.
Deeney returned the favour for Ighalo as Watford reduced the arrears six minutes into the second half, but Redmond teed up Mbokani to restore Norwich's two-goal cushion six minutes later.
"Obviously it's a severe disappointment," said Neil, the Norwich manager.
"We've known it was going to be a really difficult task. We had it in our hands three, four, five games ago, but unfortunately we came up short."
Newcastle went down despite the best efforts of manager Rafael Benitez, who was recruited in March after Steve McClaren was sacked.
The former Liverpool and Real Madrid manager sparked an upturn in both results and performances, but hopes of a dramatic escape from the bottom three were fatally undermined by last Saturday's 0-0 draw at already-relegated Aston Villa.
Benitez had a clause inserted in his contract guaranteeing that he would be released if Newcastle went down, so the St James' Park club will begin life in the Championship with a new manager.
In a statement released by Newcastle, managing director Lee Charnley said: "Tonight we are devastated at our relegation from the Premier League, as I know every single person connected to this football club will be.
"The process of scrutinising what went wrong and planning for what is to come has already begun.
"Right now, of course, Rafa and the team must prepare for the final game of the season, but after that, please rest assured we will communicate more fully with supporters."
In the day's other game, Christian Benteke's stoppage-time header earned Europa League finalists Liverpool a 1-1 draw at home to Chelsea, who had taken a 32nd-minute lead through a dazzling solo goal from belatedly in-form winger Eden Hazard.
source: www.abs-cbnnews.com
