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Manchester City effectively booked a spot in next season’s Champions League thanks to a 1-1 draw at Swansea City.
Elsewhere, Southampton beat Crystal Palace to bag Europa League football, while West Bromwich Albion held Liverpool to a draw.
Stoke City beat West Ham United, with Watford's clash against Sunderland ending all square.
News from the other final-day games can be found here.
Manchester City secured a spot in the playoff round of next season's Champions League thanks a 1-1 draw with Swansea City at the Liberty Stadium.
City hit the front early on when Kelechi Iheanacho turned home a rebound after Sergio Aguero's strike had been saved by Kristoffer Nordfeldt.
In response, Jefferson Montero headed in Angel Rangel's cross but the goal was ruled out for a push on Bacary Sagna.
Iheanacho missed a good chance to double City's advantage before, on the stroke of the interval, Andre Ayew equalised against the run of play by netting a free-kick that took a big deflection off Fernando.
A Jesus Navas cross struck the woodwork, with Iheanacho unable to convert the rebound, as the away side enjoyed the better of the third quarter.
City became a little more cautious during the closing stages but rarely came under threat from the Swans.
The single point for the departing Manuel Pellegrini's and his team will secure a place in the playoff round of next season's Champions League, unless Manchester United beat Bournemouth by 19 goals in the only remaining fixture.
Swansea end the season in 12th place.
Southampton made sure of a Europa League spot with a 4-1 victory over Crystal Palace at St Mary's.
Sadio Mane broke the deadlock close to the end of a first half that the Saints had edged, clipping home after confusion between the Palace defenders and goalkeeper Julian Speroni.
Dwight Gayle and Sullay Kaikai missed decent opportunities for the visitors before Speroni was unable to keep out substitute Graziano Pelle's header from Cucho Martina's cross.
Jason Puncheon gave Palace hope with a powerfully drilled strike following a punch out from Fraser Forster.
Southampton, however, restored a two-goal advantage when Adrian Mariappa and Pape Soure combined to foul Mane, allowing Ryan Bertrand to smash in a penalty kick.
Steven Davis iced the cake for the Saints, firing home following Martina's delivery and Pelle's knock-down.
The Saints will finish at least sixth, but would claim fifth place should Man United lose to Bournemouth in the outstanding game.
Palace, meanwhile, finish up in a disappointing 15th place.
Liverpool will have to win the Europa League to qualify for European football next season, following a 1-1 draw away to West Bromwich Albion.
A fine run early on by Jonathan Leko set up Salomon Rondon, who struck a low shot inside Adam Bogdan's near post to grab the lead for West Brom.
Jordon Ibe's first Premier League goal levelled matters, the young Londoner charging through midfield, beating Jonas Olsson and netting accurately from the edge of the penalty area.
Rondon headed an effort off the frame of the Reds' goal a few minutes into the second half and had a close-range strike blocked by Martin Skrtel, with the hosts well on top.
However, neither the Baggies nor the visitors could create much during the remainder of the clash.
West Brom finish 14th, while 'Pool take eighth place, meaning that only a victory over Sevilla in Wednesday's Europa League final will secure a spot in Europe next term.
Stoke City beat West Ham 2-1 at the Britannia Stadium, ensuring that the Hammers remain uncertain about whether they will be in next season's Europa League.
Michail Antonio's turned on Winston Reid's header down to strike the Irons ahead in a first half that the away side bossed.
However, Giannelli Imbula drilled home a low shot from 20 yards to equalise for Stoke against the run of the game.
West Ham went close several times before Mame Biram Diouf headed in Charlie Adam's corner at the near post, moving Stoke ahead.
The East Londoners could not force a leveller, allowing Stoke to finish the season in ninth place, a point ahead of Chelsea.
West Ham take seventh place, which will only be good enough for a Europa League qualifying tie if Manchester United beat Crystal Palace in the FA Cup final.
Watford and Sunderland drew 2-2 at Vicarage Road in what was essentially a celebration of both sides' participation in next season's Premier League.
A successful campaign sees the Hornets finish 12th, with Sunderland 17th, having wrapped up their escape from relegation with a midweek win over Everton.
Jack Rodwell fired in from a low cross by Jeremain Lens to give the Black Cats the lead.
Sebastian Prodl headed Watford level just after the interval but Lens quickly drove Sunderland back in front.
However, Troy Deeney's penalty kick ensured parity, following John O'Shea's foul on Jose Manuel Jurado.
Watford dominated the closing stages, although Sunderland had two goals disallowed on the day, as Quique Sanchez Flores's final game in charge of the Hornets ended all square.