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Sunderland striker Danny Graham scored his first goal in the Premier League since December 2013.
Also on the scoresheet for the Black Cats was Jermain Defoe, who found the net for the fourth time since his January arrival from Toronto.
Graham's goal was the first that Everton had conceded at Goodison Park in seven hours.
Aston Villa 1-0 West Ham United
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The clean sheet was the first that Villa have kept at home in the Premier League since January 1.
They won thanks to a goal from Tom Cleverley, who found the net for the third consecutive match.
As for West Ham, they have scored just four goals away from home in 2015.
Hull City 0-1 Burnley
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Steve Bruce's Hull are still without a Premier League win in the month of May, drawing three times and losing eight.
Burnley's first win in seven matches was not enough to stop them from suffering relegation.
Match-winner and reported Liverpool target Danny Ings ended a run of 10 Premier League matches without a goal.
Leicester City 2-0 Southampton
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Winger Riyad Mahrez found the net for the first time in the Premier League since December 28.
The Foxes have now collected 18 points from the last 21 available.
In contrast, Southampton have failed to win any of their last four matches, losing on three occasions.
Newcastle United 1-1 West Bromwich Albion
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West Brom striker Victor Anichebe scored his fifth goal against Newcastle, which is more than he has scored against any other Premier League side.
Newcastle have now gone nine Premier League games without a win - their longest run since 2009.
Under John Carver, the Magpies have won just one of their nine matches at St James' Park.
Stoke City 3-0 Tottenham Hotspur
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Tottenham's poor run of form continued, having now won only one of their last six outings.
It is the first time that Spurs have lost both league matches in one season to the Potters.
Mame Biram Diouf has scored in four of Stoke's last five matches at the Britannia Stadium.
Crystal Palace 1-2 Manchester United
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All seven of Marouane Fellaini's goals for Manchester United have been scored this season.
Meanwhile, winger Ashley Young claimed his fourth assist of the season, which is the same return he managed in the previous two campaigns combined.
Palace have won only three of the nine matches that Alan Pardew has taken charge of at Selhurst Park.
Manchester City 6-0 Queens Park Rangers
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The 67 goals that relegated QPR have conceded so far is the most that they have ever shipped in a Premier League season.
Leading goalscorer Sergio Aguero's hat-trick took his tally for the campaign to 25, which is five more than second-placed Harry Kane.
The Argentine has scored six goals in his five appearances against the West Londoners.
Chelsea 1-1 Liverpool
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Chelsea skipper John Terry's 39th goal makes him the record-scoring Premier League defender.
Blues debutant midfielder Ruben Loftus-Cheek completed all 27 of his passes before he was substituted.
Steven Gerrard's last two Premier League goals have been scored with his head.
Arsenal 0-1 Swansea City
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Arsene Wenger named the same Arsenal starting lineup for the fifth Premier League game running - the first time he's ever done so since taking charge of the club.
The Swans have become only the third team in Premier League history to have completed the double over both Arsenal and Man United in the same season.
French striker Bafetimbi Gomis has found the net four times in his last four outings.
