The game was manager David Moyes's last in charge of the Blues at Goodison Park before he leaves the Merseyside outfit for Manchester United in the summer.
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Two minutes later Everton could have doubled their advantage when Darron Gibson tried his luck from outside the West Ham box, but failed to keep his effort on target.
West Ham got back into the game thereafter as Mohamed Diame and Kevin Nolan threatened to breach Tim Howard's citadel, before Everton defender Sylvain Distin headed Mirallas's corner just wide in the 38th minute.
The hosts continued to have the upper hand in the second half and went 2-0 ahead on the hour mark, Mirallas scoring his second goal of the afternoon after collecting Gibson's pass and getting it in with the deflection off of James Collins.
That proved to be the final goal at Goodison as Everton cruised to all three points.