There is now only one more spot in the bottom three to be filled after Fulham and Cardiff lost to Stoke City and Newcastle United respectively, and Norwich City need to avoid defeat against Chelsea on Sunday to keep their hopes of avoiding that fate alive.
West Ham United secured their safety by beating Tottenham Hotspur and Aston Villa can breathe easily after seeing off Hull City, who would also need an unlikely set of results now to drag them down.
Elsewhere, Southampton beat Swansea City, but the key result of the day which had ramifications for all the sides in the bottom eight was Sunderland's success at Old Trafford.
Ryan Giggs earned a lot of praise after winning his first match in interim charge of Manchester United last weekend, but he was powerless to prevent Sunderland's remarkable resurgence from continuing with a 1-0 away win.
Connor Wickham has scored some crucial goals over the past few weeks but he turned provider on 30 minutes to help Sunderland make the crucial breakthrough, swinging over a cross for Sebastian Larsson to steer neatly home.
United tried to battle back after the break and had some dangerous moments, but it was Sunderland who came closest to scoring again as Emanuele Giaccherini and Fabio Borini both hit the woodwork.
Sunderland are now three points clear of Norwich and both sides have two games left to play, but Gus Poyet's side appear to have the kinder fixtures and boast a far superior goal difference, while the momentum is definitely with them after a first league win at Old Trafford since 1968.
Fulham's 13-year stay in the Premier League will come to an end when the season finishes after they were thrashed 4-1 at Stoke in their penultimate game, leaving them an impossible four points from safety.
Peter Odemwingie opened the scoring from close range for Stoke in the 39th minute after Stephen Ireland's shot was deflected onto the crossbar by Dan Burn.
Marko Arnautovic doubled Stoke's advantage nine minutes into the second half when he galloped forward into yards of space following a swift break and slotted home a composed finish from Odemwingie's pass.
Another rapid counter-attack resulted in Oussama Assaidi turning in Stoke's third on 73 minutes, before Kieran Richardson slotted home from Darren Bent's pass to give Fulham the smallest of lifelines with ten minutes to go.
However, Fulham's fate was sealed and a miserable afternoon ended with Jon Walters completing the rout for Stoke.
Cardiff will join Fulham in the Championship next season after going down 3-0 to Newcastle at St James' Park.
Newcastle had lost their previous six Premier League matches but they started strongly and Shola Ameobi headed them in front on 18 minutes from Moussa Sissokho's cross to the back post.
The match situation did not stop some Newcastle fans from staging a walk-out in protest against Alan Pardew and Mike Ashley midway through the second half, but Newcastle held on to triumph as Loic Remy and Steven Taylor scored goals from close range in the final three minutes to add gloss to the scoreline.
Manchester City took the opportunity to climb to the top of the table with a thrilling 3-2 win at Goodison Park which extinguished Everton's hopes of finishing fourth.
Ross Barkley fired Everton in front with a spectacular curling strike after 11 minutes but Sergio Aguero drew City level on 22 minutes after being played in by Yaya Toure.
Edin Dzeko headed in James Milner's cross after 43 minutes and then put City 3-1 ahead early in the second half from Samir Nasri's cross.
Leighton Baines' delivery was nodded in by Romelu Lukaku with 25 minutes still to play to give Everton hope, but City held on to go above Liverpool on goal difference with a couple of home games to follow.
Aston Villa overcame Hull 3-1 in a clash between two sides who started the weekend still in with a chance of going down, but following results elsewhere Villa are now all-but safe and Hull have a five-point cushion.
Villa's nerves were settled by a goal inside the opening minute as Gabriel Agbonlahor delivered a cross from the left and Marc Albrighton, after missing the ball with his initial attempt at a first-time shot, laid it off for Ashley Westwood to pick his spot in the bottom corner.
A cleverly-worked free-kick routine after 28 minutes led to Hull's equaliser as Tom Huddlestone played the set-piece quickly to Liam Rosenior and, as the ball was fizzed across the six-yard box, substitute Jordan Bowery could do nothing but turn it into his own net.
Andreas Weimann restored Villa's lead when he nodded in after a desperate scramble in the area, and he then made it 3-1 with a much more stylish goal as he glanced a header past Steve Harper from Bowery's inviting cross.
In the day's early game, West Ham ended a run of four defeats with a 2-0 victory over Tottenham to raise their points tally to 40 and guarantee their top-flight status for another season.
Sam Allardyce has grown frustrated by criticism from fans in recent times but the Upton Park faithful were celebrating after 25 minutes when Stewart Downing was pulled back outside the area by last man Younes Kaboul and the Spurs defender was shown a straight red card.
West Ham took the lead just a couple of minutes later when Andy Carroll's header took a crucial deflection off Harry Kane to loop over Hugo Lloris.
Downing added a second with a free-kick just before half-time which was driven straight through a disappointing defensive wall, and Spurs could not find a way back into the match with ten men as their hopes of European qualification suffered a blow.
Southampton edged out Swansea 1-0 in an uninspiring contest at the Liberty Stadium.
The Swans had ensured their safety last week and there was an end-of-season feel to the match, but Southampton snatched all three points to make certain of finishing eighth when Rickie Lambert bundled the ball over the line in the 90th minute.