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The early goal forced an immediate reaction from Everton and the visitors were close to equalising just moments later when Marouane Fellaini found space but his low strike was pushed away by Mark Schwarzer.
Fulham rarely troubled the Everton defence after the early breakthrough but the visitors continued to be wasteful in front of goal, with Leon Osman failing to stretch Schwarzer with his curled effort from the edge of the area.
The hosts somehow managed to keep out their opponents in the first half but Everton's dominance continued after the break when Leighton Baines delivered a free kick that just needed a touch to beat Schwarzer.
David Moyes finally saw his side equalise after 55 minutes as Kevin Mirallas beat a couple of defenders before delivering a perfect cross to allow Fellaini to smash the ball in from close range.
Fulham couldn't find an answer to Everton's movement in attack and the away side took the lead with 18 minutes remaining when Fellaini brought the ball down and then fired his shot low past Schwarzer.
Martin Jol's side almost replied in the closing stages but Brede Hangeland headed over from close range.
However, Everton failed to extend their lead and were punished in the final minute as Sidwell tapped home from close range to earn a point for Fulham.