The Belgium striker scored both goals to provide a fitting send-off for his boss - heading for Manchester United in the summer - after he has stood on the touchline in the blue corner of Merseyside for 11 years.
The sixth successive home league victory against a West Ham side who seemed to have no intention of spoiling the party also guaranteed Everton finished above Liverpool for the second successive season in and the first time in the top flight since 1937.
The players started as if they wanted to give their manager the performance his excellent long-term work deserved, taking only six minutes to break the deadlock.
In a crisp passing move, the ball was moved in from the left touchline from Leighton Baines to Marouane Fellaini, the two players most likely to be coveted by Moyes for his new dawn at United, and then on to Steven Pienaar who teed up Mirallas to drill home from the edge of the area.
Everton, as has so often been the story this season, should have capitalised on their advantage as a Darron Gibson chest-trap and volley dipped onto the roof of the net while Victor Anichebe fired straight into the groin of goalkeeper Jussi Jaaskelainen from close range.
The visitors looked like a side safe in mid-table as their only two chances came from captain Kevin Nolan in the last 15 minutes of the first half.
Goodison turned up the volume in the second half to give Moyes an idea of what he was leaving behind but the manager was more concerned with no-one following in Leon Osman's cross-shot through the six-yard box from Baines' delivery.
But the Scot was all smiles and double fist-pumps on the hour when Pienaar released Mirallas to run at the defence and his shot deflected up off James Collins and past Jaaskelainen.
Even West Ham's on-loan Liverpool striker Andy Carroll could not but a dampener on proceedings, hitting the post with a late header.
Source: PA
Source: PA