Marouane Fellaini headed in a late winner as Manchester United beat Crystal Palace 2-1 on Saturday to move within touching distance of a return to the Uefa Champions League.
United will be guaranteed a top-four finish if Liverpool lose to newly-crowned champions Chelsea on Sunday.
Fellaini rose at the back post to nod in a cross by Ashley Young in the 78th minute as United ended a run of three straight defeats. Juan Mata’s 19th-minute penalty for United was canceled out by a curling free kick from Jason Puncheon in the 57th.
United were indebted to goalkeeper David De Gea, who made great saves either side of Fellaini’s goal.
United are seven points ahead of fifth-place Liverpool, who have three games left.
Burnley were relegated from the Premier League on Saturday despite winning 1-0 at fellow strugglers Hull City, which left them seven points from safety with only two games remaining.
Promoted from the Championship last season, Burnley gave themselves hope of avoiding the drop by stunning defending champions Manchester City 1-0 in March, but they subsequently went six games without winning, losing five.
Danny Ings stopped the rot with a 62nd-minute winner at the KC Stadium, crashing home left-footed after Hull failed to clear a corner to end a 1,036-minute goal drought for Sean Dyche’s side, but it was to no avail.
“The lads showed unbelievable fight and spirit today,” Ings said. “I thought we were phenomenal and were on top from start to finish, but it has come a little too late.
“The lads have fought until the end, and I’m extremely proud of them. Now we are going to play the last two games with pride and hope to finish strongly.”
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It is the second time Burnley have been relegated from the Premier League a season after securing promotion, having previously met the same fate in 2009-10.
The result dragged Hull into the relegation zone and left Steve Bruce’s side two points from safety ahead of their thorny final fixtures away to Tottenham Hotspur and at home to Manchester United.
Burnley’s victory lifted them above Queens Park Rangers, who are on the brink of slipping into the second tier as they are nine points adrift of safety ahead of their trip to Manchester City on Sunday.
Newcastle United skidded closer to the relegation zone despite halting an eight-game losing streak -- the club’s worst sequence since 1977 -- by drawing 1-1 at home to West Bromwich Albion.
Victor Anichebe headed West Brom in front in the 32nd minute, but Newcastle striker Ayoze Perez equalised nine minutes later by gathering a free-kick from Ryan Taylor and finding the bottom-right corner with his left foot.
Newcastle are now one place and two points above the bottom three, level on points with local rivals Sunderland, who won 2-0 at Everton courtesy of scrappy second-half efforts from Danny Graham and Jermain Defoe.
It was Sunderland’s first away win since a 1-0 success at Newcastle on December 21.
Leicester City’s stunning resurgence continued as they made it six wins from seven games with a 2-0 victory at home to Southampton that left Nigel Pearson’s side three points above danger.
Algerian winger Riyad Mahrez scored both goals in the first 19 minutes, first drilling home left-footed from the edge of the box and then slamming home a left-wing cross from Jamie Vardy.
FA Cup finalists Aston Villa also took a step closer to safety by beating West Ham United 1-0 at Villa Park, with on-loan Manchester United midfielder Tom Cleverley scoring his third goal in three games.
Like Southampton, Tottenham do not seem particularly motivated by the prospect of qualifying for the Europa League, but they remain a point above Ronald Koeman’s men in sixth place despite crashing 3-0 at Stoke City.
Charlie Adam headed Stoke in front in the 21st minute and Steven Nzonzi exploited a mix-up between Hugo Lloris and Eric Dier to make it 2-0 before Mame Biram Diouf added a late third.
Spurs defender Vlad Chiriches was sent off early in the second period after receiving a second yellow card.
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