Aston Villa survived the drop thanks to a 1-1 draw at West Ham United which moved them to 35 points, one more than Bournemouth, who go down despite a 3-1 win at Everton, and Watford who lost 3-2 at Arsenal.
A penalty from Bruno Fernandes and late goal from Jesse Lingard gave United a 2-0 win at Leicester City, who missed out on the Champions League by finishing fifth and will have to settle for a spot in the Europa League.
Chelsea ended fourth after a 2-0 home win over Wolverhampton Wanderers thanks to goals from Mason Mount and Olivier Giroud.
Jose Mourinho’s Tottenham Hotspur, who drew 1-1 at Crystal Palace, are also in the Europa League after finishing sixth.
It was a nervous display from United against an injury-ravaged Leicester but they ended the campaign unbeaten in 14 Premier League games and with their mission of regaining a place in the Champions League accomplished.
Draws against Southampton and West Ham had left United needing at least a point at the King Power stadium and while they again looked tired in a nervous display, Ole Gunnar Solksjaer’s side got the job done.
Fernandes, who was central to United’s rise in the second half of the season, put United ahead in the 71st minute after Anthony Martial was brought down in the box.
The win was wrapped up when substitute Lingard robbed Leicester keeper Kasper Schmeichel on the edge of the area and slotted into the unguarded goal.
“We looked a bit tired and leggy, I think everyone could see. It wasn’t a classic,” said Solskjaer who will now hope the club back him in the transfer market.
“The club know what I feel we need and we are going to try to pursue it and let’s see where we end up,” the Norwegian added.
Chelsea also needed just a point at Stamford Bridge to guarantee a place in Europe’s top club competition and a wonderful free kick from Mount and confident finish from Giroud put them two goals up by the break — a lead Wolves never threatened to damage.
There was tension at the bottom, though, with Villa taking the lead at West Ham thanks to Jack Grealish in the 84th minute but the hosts hit back quickly through Andriy Yarmolenko, leaving Dean Smith’s side anxiously waiting on Watford’s result.
Watford had been 3-0 down at Arsenal but a Troy Deeney penalty and Danny Welbeck goal gave the Hornets hope.
It was not to be, however, for the club who sacked manager Nigel Pearson with two games remaining.
“This is a reflection of the whole year. Not quite been good enough,” said skipper Deeney.
“It is heartbreaking for the people who work behind the scenes and we feel sorry for them. As a club and as a community we will go again. Ultimately we have not been good enough, there is no point dancing around it,” he added.