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Two first-half goals put the Toffees on their way, before Mikel Arteta's own goal wrapped up the points to leave the Gunners hanging on to the final Champions League spot.
The hosts almost got off to a dream start when Leon Osman struck a volley from the corner of the area, but the ball swerved just wide of the target to the relief of Wojciech Szczesny.
Roberto Martinez's men increased the pressure further in search of a second, and following a succession of near-misses, Lukaku found a way through to make it 2-0.
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The Belgian international collected Kevin Mirallas's pass on the right-hand side, before cutting inside and finishing low into the far corner to put the Toffees into a commanding position 10 minutes before the interval.
Arsenal came close to pulling one back through Lukas Podolski when his mishit shot bobbled up and nearly over Tim Howard, but the American keeper managed acrobatically tip the ball over to retain his side's two-goal cushion.
Olivier Giroud almost capitalised on some sloppy Everton play shortly after the restart, with the striker being gifted an opportunity by Mirallas to find the net, although Howard was quick to close him down to save his teammate's blushes.
Any hopes Arsenal had of finding a route back into the game were soon over, however, when Bacary Sagna was dispossessed leaving Mirallas to charge up field and play in Naismith.
The winger was initially beaten to the ball by Szczesny, but Arteta was unfortunate to be in the path of the clearance which he ended up sliding into his own net.
Alex Oxlade-Chamberlain came within inches of pulling back a consolation for Arsene Wenger's charges with a strike from 25 yards, only for the ball to come back off the bar, while Yaya Sanogo's 'goal' was ruled out for being offside in the dying seconds.