The Belgium international struck from an electrifyingly quick counter-attack in the first half which left defenders trailing in his wake.
His sixth goal of the season, although only his third in 18 league appearances, was enough to ensure David Moyes' side built on their victory over champions Manchester City before the international break.
Stoke should have gone ahead within three minutes. Ryan Shotton's long throw picked out Robert Huth, whose header was well saved by Tim Howard - but Jon Walters crashed the follow-up against the crossbar from close range.
Aside from two Nikica Jelavic headers, one straight at Asmir Begovic and the other deflected wide by Huth, Everton produced little in the opening 25 minutes.
Goalkeeper Tim Howard was returning to the starting line-up after recovering from two broken bones in his back but no-one expected him to provide the assist for the opening goal in the 28th minute. Marc Wilson's cross from a short free-kick was punched clear by the American and landed at the feet of Mirallas, the Everton player highest up the pitch.
Although the Belgium international gave Steven Nzonzi a chance to nick the ball in the centre-circle the midfielder could not accept the invitation, got in a tangle and his opponent raced clear.
There are few who can match Mirallas in full flow and although Stoke scrambled to get back the £6million summer signing could not be caught, clipping a shot which Begovic got a hand to but could not stop.
With his confidence boosted the former Olympiacos forward then thrashed a shot across Begovic's goal and wide of the far post from the right corner of the penalty area.
In the second half Huth planted a header wide from Glenn Whelan's free-kick when he should have hit the target - and therein lies Stoke's problem. When their considerable strengths, the physicality which grinds teams down and the aerial ability which often causes opponents so many problems, fail them there is nothing else in reserve.
Source: PA
Source: PA