The midfielder netted his sixth goal of the season before turning provider for the Croatia international, to extend his side's unbeaten run in the Premier League to nine games
The visitors took the lead in first-half stoppage time, with summer signing Adam Johnson scoring his first goal for the club.
The home side reacted well, piling the pressure on the Black Cats’ defence throughout the second half and, with 14 minutes to play, Marouane Fellaini produced two moments of magic to hand his side victory.
The Belgian scored his sixth goal of the season for the Toffees, before turning provider for Nikica Jelavic just seconds late.
The hosts named an unchanged side, with Seamus Coleman continuing at right-back in the absence of Tony Hibbert (calf) - who is expected to return next weekend.
For the visitors, Danny Rose was given the nod ahead of Phil Bardsley in defence, while James McClean came into midfield at the expense of Lee Cattermole who was ruled out late in the afternoon with a hamstring complaint.
There was action from the off at Goodison Park, as no sooner had Jelavic had a soft penalty shout turned down at one end, Stephane Sessegnon raced in behind Coleman and tested Tim Howard at the other.
Moments later, Sessegnon set Steven Fletcher away with a defence-splitting pass, but the visitors’ top goalscorer fired narrowly wide of the target.
The hosts then began to find their rhythm, and after impressively completing over thirty passes around the edge of the Sunderland penalty area, Jelavic and Mirallas combined well to create a chance for the latter, but James McClean did well to intercept.
The Toffees continued to threaten the Black Cats goal, as Simon Mignolet was tested three times in quick succession - first by Jelavic and then twice by Steven Pienaar - but the Belgian was equal to all that came his way and kept the scores level.
But, soon after, Everton were sucker punched by Sunderland taking the lead against the run of play.
A sloppy clearance from a corner found Carlos Cuellar on the edge of the penalty area. The Spaniard lofted the ball back into the Everton penalty area towards Johnson, who managed to evade Leon Osman and slot the ball past an on-rushing Howard.
The home side came out of the break revitalised, with Gary Naismith and John Heitinga both going close after two fantastic deliveries from Leighton Baines on the left.
In spite of concerted pressure from the hosts, the Sunderland duo of Jack Colback and Sebastien Larsson put in a dogged display in midfield, with a number of crucial, last-ditch tackles to shield their defence.
But with 14 minutes to play, top-scorer Fellaini found himself in space on the edge of the visitors’ penalty area. Cuellar was slow to close the Belgian down, allowing him to turn and neatly slot the ball past Mignolet.
And, seconds later, Fellaini received the ball on the edge of the box again, but this time performed a delightful one-touch flick through the legs of John O’Shea to find Jelavic, who made no mistake from six yards.