In truth the home side should have won more comfortably, with Jussi Jaaskelainen and James Collins to thank for keeping the home side at bay.
It was less than six minutes until Everton took the lead. Leighton Baines drove down the left wing, laying the ball back to Marouane Fellaini. Fellaini in turn played in Steven Pienaar, who unselfishly passed to Mirallas rather than shoot himself, and the Belgian's side-foot gave Jaaskelainen no chance.
The early goal set the tone for the first half, with Fellaini and Leon Osman acting as the pivot from which full-backs and wingers could operate. Mirallas in particular looked dangerous, and almost had another opportunity to score when the otherwise excellent Collins was involved in a mix-up with his goalkeeper.
Halfway through the first half and Jaaskelainen again thwarted Everton from extending their lead. Fellaini's shot struck a defender and fell to Victor Anichebe. The striker looked offside (although replays showed not) but Finnish keeper smothered the save from close range.
Five minutes later and Tim Howard had his first save to make when Kevin Nolan struck a left-footed volley from the edge of the area. The American had to be a full stretch to keep out the effort.
The second half started as the first ended, despite the introduction of Joe and Carlton Cole by Sam Allardyce. Seamus Coleman twice played dangerous balls into the box which found only space, before a Baines cross was met by Osman, who could only fire across goal.
On the hour mark, Everton's lead was doubled. A fine ball from Darron Gibson found Mirallas, and the Belgian fired at goal from 20 yards. The ball took a horrible deflection off Collins and looped over his goalkeeper.
The home side were in danger of running riot, and within ten minutes had four more clear cut chances. Collins made another two fine blocks, whilst Jaaskelainen made a fantastic save from Osman
With 20 minutes left whilst Anichebe headed harmlessly wide when presented with a great opportunity to score, and it was the striker's last action, replaced by Nikica Jelavic.
Howard was forced into making another save, diving low and parrying out Joe Cole's effort. With ten minutes to play, but that was West Ham's only effort of note during the second half until Andy Carroll headed against a post with four minutes to play.
As the final whistle blew, the Goodison faithful rose to acclaim their departing hero, and Moyes deservedly took the plaudits with misty eyes.