David Moyes' men remain in contention for a top-four Premier League finish and are now just four points adrift of Chelsea in the battle for UEFA Champions League football next season.
Stoke, meanwhile, have now lost eight times in their last 11 Premier League outings and have slipped to 13th in the table, only four points above the increasingly congested battle against the drop.
Everton welcomed goalkeeper Tim Howard and Phil Jagielka back into their starting line-up and they were joined in the team by Nikica Jelavic as Moyes made three changes and opted for a 3-5-2 formation with Marouane Fellaini and Steven Pienaar suspended.
Stoke selected centre-back Robert Huth after his three-game ban as he replaced the injured Matt Etherington, with the German helping the Potters go close to opening the scoring with just three minutes gone.
Ryan Shotton's long throw into the box was met by Huth's head with Howard palming it away as he threw himself to his left, with Jon Walters, who lined up in midfield in Tony Pulis' 4-4-2 formation, scrambling his follow-up effort against the underside of the bar.
Leighton Baines had come close to providing Jelavic with the ammunition to test Stoke goalkeeper Asmir Begovic on more than one occasion before Everton took the lead courtesy of a majestic solo effort from Mirallas.
The Belgian collected possession deep inside his own half as Howard fisted away a Stoke free-kick, wriggling away from the attentions of Steven N'Zonzi before switching on the after-burners and embarrassing Geoff Cameron as he turned the last defender inside and out before slotting under Begovic.
Howard had to be alert again as the interval approached, plunging away to his left to push away a stinging effort from Walters from outside the box with Stoke remaining a threat to Everton's back three.
Stoke continued to create decent openings after the interval with Huth's downward header from Whelan's free-kick in the 57th minute whistling the wrong side of Howard's upright, while Shotton could not control his chance at the far post soon after as Marc Wilson's cross flicked off Distin.
Everton could have extended their lead in the 63rd minute when Jelavic's far post header from Darron Gibson's cross went straight at Begovic, before the Croatia striker brought a finger-tip save out of the Potters keeper with a free-kick from the edge of the box.
Cameron Jerome flashed over the bar after Shotton's back-heel before deep into added time substitute Charlie Adam stung Howard's palms at the near post after a free-kick had incorrectly been awarded following Leon Osman's challenge on the former Liverpool player.