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Tom Cleverley snatched victory at Newcastle with the last touch of the game as Everton finally made the pressure count.
The midfielder looped home a 93rd-minute header to finally beat Magpies keeper Rob Elliot, who had made six fine saves, including two from man of the moment Romelu Lukaku as he looked to score in an eighth successive league game.
It was just a second Barclays Premier League win in seven attempts for the Toffees, and while it was deserved on the balance of play, Newcastle will curse their luck having got so close to riding out the storm and dragging themselves back out of the relegation zone.
The visitors took charge from the off, with Seamus Coleman and Aaron Lennon down the right and Leighton Baines and Kevin Mirallas down the left causing all kinds of problems and Ross Barkley and Cleverley prompting from midfield.
Lukaku was the main beneficiary of all the hard work behind him and Magpies skipper Fabricio Coloccini had his hands full with the Belgian throughout.
But it was Mirallas who came closest to opening the scoring in the early stages, forcing Elliot to block with his legs at the far post from Coleman's seventh-minute cross and then get down to claim his snapshot from the edge of the penalty area 12 minutes later.
Elliot came to the rescue once again with 28 minutes gone when he kept out Lukaku's low drive with his legs after the striker had been played in by Coleman, seconds after Coloccini appeared fortunate to get away with a tug on the frontman's shirt as he burst through inside the box.
Newcastle responded and striker Aleksandar Mitrovic glanced a header wide from Moussa Sissoko's deflected 30th-minute cross after earlier failing to make meaningful contact with Daryl Janmaat's ball in.
But it was Georginio Wijnaldum who got on the end of the home's side's best move of the half eight minutes before the break, heading over the angle of post and bar from another Sissoko cross.
Lukaku was just inches away from meeting Coleman's near-post cross two minutes after the restart, and Elliot had to block the same man's effort with his legs after the striker had powered his way past central defender Chancel Mbemba and then get down to repel Lennon's skidding 59th-minute effort.
The game became increasingly open as Newcastle sought to hit Everton on the counter and Ayoze Perez fired just wide on the turn with 19 minutes remaining, but Mitrovic should have scored two minutes later when he met Sissoko's cross with a powerful header but steered it wide.
Wijnaldum called upon Everton keeper Tim Howard for the first time with a header from Jack Colback's cross which the American managed to beat away, but Elliot had to claw away Ramiro Funes Mori's effort at the other end.
But Elliot was finally beaten at the death when he could only punch a corner to Cleverley, who looped a header home to claim the points.
TWEET OF THE MATCH
@GaryLineker: "What an absolute delight it is to watch an English central defender play in such a relaxed manner on the ball. John Stones is a class act."
Rob Elliot: 8
Daryl Janmaat: 6
Chancel Mbemba: 7
Fabricio Coloccini: 5
Paul Dummett: 5
Vurnon Anita: 6
Jack Colback: 6
Moussa Sissoko: 7
Ayoze Perez (Thauvin, 87 mins): 6
Georginio Wijnaldum: 7
Aleksandar Mitrovic: 6
Florian Thauvin: 6
Tim Howard: 6
Seamus Coleman: 7
John Stones: 7
Ramiro Funes Mori: 6
Leighton Baines: 7
Gareth Barry (Capt): 7
Tom Cleverley: 7
Aaron Lennon (Besic, 69 mins): 6
Ross Barkley (Deulofeu, 69 mins): 7
Kevin Mirallas (Osman, 85 mins): 7
Romelu Lukaku: 8
Gerard Deulofeu: 7
Muhamed Besic: 6
Leon Osman: 6
Newcastle keeper Rob Elliot produced six vital saves to keep Everton at bay for all but the final few seconds on a night when the visitors might have won much more comfortably had it not been for his efforts.
MOMENT OF THE MATCH
Elliot reacted superbly to claw out Aaron Lennon's skidding 59th-minute after as it sped through a crowded penalty area.
VIEW FROM THE BENCH
Everton boss Roberto Martinez will have been frustrated at the number of chances which went begging, but delighted with the football his side played to pave the way for victory. Newcastle head coach Steve McClaren saw his side scrap all the way and gradually work their way into the game, but will have been concerned at how easy the visitors found the going before the break.
MOAN OF THE MATCH
Newcastle's inability to find one of their own men from a set-piece with both Ayoze Perez and Jack Colback guilty.
WHO'S UP NEXT
West Brom v Newcastle (Barclays Premier League. Monday).
Everton v Stoke (Barclays Premier League, Monday).