A remarkable number of Jussi Jaaskelainen saves and James Collins blocks saw the scoreline kept respectable as the Toffees romped, though the latter was unfortunate to see Mirallas' second effort take a hefty deflection of him to snuff out any hopes of an unlikely Hammers revival.
Set to end his 11-year tenure at Everton by replacing Sir Alex Ferguson at Manchester United next season, Moyes leave the club in good standing, with the win guaranteeing a sixth-place finish in the Premier League while also confirming a spot in next season’s Europa League.
Phil Neville’s knee injury meant that the soon-to-depart club captain was unable to join Moyes in a last hurrah in front of the Toffees faithful, while Sam Allardyce made a solitary change from the side that drew 0-0 Newcastle last weekend, with Ricardo Vaz Te replaced by Jack Collison.
Acting on a raucous atmosphere inside the stadium, Moyes’ side powered ahead after just five minutes. Sylvain Distin should have been the man to grab the opener, though he struck wide from just six yards out moments before Mirallas - teed up by Steven Pienaar - passed beyond Jussi Jaaskelainen in clinical fashion.
The Hammers struggled to come to terms with the opposition's inventive surges forward, and an Everton onslaught soon ensued. Victor Anichebe, Darron Gibson and Leon Osman all went close with decent efforts, as Jaaskelainen did well to block Anichebe's close-range smash.
Kevin Nolan endeavored to drag West Ham back into the game before the interval but was denied first by Tim Howard's plunging save on the half hour mark, then by his own inaccuracy when he fired wide on the turn after Phil Jagielka's hashed clearance.
Allardyce's tactical rethink - a halftime reversion to 4-4-2 with Carlton Cole replacing Mohamed Diame - failed to prevent Everton regaining control, and within minutes Seamus Coleman saw his effort blocked, while Osman squared Leighton Baines' deep delivery across an empty six-yard box.
Mirallas angled narrowly over with his left-foot after skinning James Collins but then had the center half to thank for doubling his tally shortly after, as his effort cannoned off Collins to leave both Jaaskelainen wrong-footed and Everton two goals up.
Still the chances flowed for the Toffees, though they found a wall in the shape of Jaaskenlainen, who thwarted Anichebe, Pienaar and Osman twice, with saves ranging from the improbable to the miraculous.
From there on Everton saw out the rest of the game in total comfort, in the process securing a sixth-placed Premier League finish that sees the club once again better the efforts of rival Liverpool's domestic campaign, while West Ham enters its last game of the season battling for a top-half finish on its return to the top flight.
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