The two own goals scored by Everton players took their tally for the Premier League era to 43 - more than any other club.
It also meant that the Toffees have failed to keep a clean sheet in nine of their previous 11 encounters.
Meanwhile, the victory was Southampton's first over Everton in nine attempts.
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A Venezuelan footballer had never previously scored in the Premier League prior to Fulham defender Fernando Amorebieta finding the back of the net.
Also, Fulham's other goalscorer, Ashkan Dejagah, has scored three times in his last four outings at Craven Cottage.
Hull striker Nikica Jelavic ended a drought of seven matches without a goal against the West Londoners.
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Stoke City 0-1 Tottenham Hotspur
All three of Danny Rose's Premier League goals have been scored in the month of April.
It was also his second goal in the top flight for Spurs - the first was scored four years ago against Arsenal.
The defeat was Stoke's first at the Britannia Stadium in six outings.
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Swansea City 4-1 Aston Villa
All of Jonjo Shelvey's last five goals have been scored from outside of the penalty area.
Also among the scorers was Swansea centre-forward Wilfried Bony, who has now scored in six of his last seven appearances.
Villa are now without a victory from their last six matches.
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West Bromwich Albion 1-0 West Ham United
The defeated Hammers have lost seven of their previous nine Premier League fixtures.
As for the hosting Baggies, this was their first victory at The Hawthorns since New Years' Day.
The winning goal was scored by Saido Berahino, who has scored twice against West Ham this season, which has earned his side four points.
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Manchester United 4-0 Norwich City
Juan Mata came off the bench to score two goals, taking his tally to four in his last five matches for Man United.
One of those was scored with his head, which is the first headed goal that the Spaniard has scored in the Premier League.
As for Norwich, they have lost eight away matches on the trot for the first time in their history.
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Sunderland 4-0 Cardiff City
Having scored once in his previous 37 Premier League appearances, Connor Wickham has now scored five in his last three.
The 4-0 scoreline was Sunderland's biggest winning home margin since September 2011 when they defeated Stoke City.
Meanwhile, Spanish defender Juan Cala was the first Cardiff player to receive a red card this season.
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Liverpool 0-2 Chelsea
Chelsea manager Jose Mourinho has lost just one of his nine Premier League encounters with Liverpool.
The win meant that the Blues have taken 16 points from a possible 18 against the three other clubs in the top four.
Liverpool captain Steven Gerrard had nine shots over the course of the 90 minutes, which is only two fewer than the entire Chelsea team.
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Crystal Palace 0-2 Manchester City
The defeat brought an end to Palace's five-match winning streak.
It was also their fourth defeat in a row against City, who recorded their first clean sheet in six outings.
Following his return from injury, Yaya Toure has scored six times in as many Premier League matches.
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Arsenal 3-0 Newcastle United
The visiting Magpies have lost 10 of their last 13 Premier League games, yet they still remain in the top half.
Alan Pardew's side have now conceded 57 goals, which is the same number as their struggling rivals Sunderland.
The result ensured that Arsenal will finish above North London rivals Tottenham for the 19th successive campaign.
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