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.
Everton began the day six points outside Champions League qualification but by the final whistle they were just four behind fourth-placed Chelsea, who slipped up at Southampton.
Their quest to play among Europe's elite for only the second time remains a possibility and the first half of next month is crucial to their hopes as they face Spurs and, more importantly, Arsenal - the team who stand between them and the top four.
Stoke, however, are looking over their shoulders with increasing anxiety as one win in 12 league matches has brought the bottom of the table into closer focus than manager Tony Pulis would have expected after finishing 2012 unbeaten in 10 games.
Goals, or a lack of them, are the Potters boss's concern with just six in 11 league games, which has contributed to a Premier League worst tally of 14 matches this season without scoring.
With Phil Jagielka returning from a three-match absence with an ankle injury to captain the side Moyes, without the suspended key duo of Marouane Fellaini and Steven Pienaar, unusually opted for a back three.
It took a while for the side to settle in this new and unfamiliar formation and as a result 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 Robert Huth, Everton produced little in the opening 25 minutes.
Goalkeeper 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.
The best Stoke could muster in response was Walters' curling shot which was palmed away acrobatically but theatrically by Howard.
In the second half Huth planted a header wide from 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.
Shotton also squandered a more difficult effort, Wilson's inswinging cross deflecting off Sylvain Distin's back, from a similar position.
Ryan Shawcross' rugby tackle on Jelavic to prevent the Croatian running clean through on to Leon Osman's pass earned the Stoke captain a booking and Everton a free-kick on the edge of the area but Leighton Baines blasted into the wall.
Fortunately for the hosts this was not a day when they needed the cushion of the extra goal as Stoke surprisingly hardly tested Howard and his recently-repaired broken back.