© PA Photos
The visitors almost made a perfect start to the match when Jonathan Walters came close after just five minutes. The Irishman had the goal at his mercy from six yards out, but clattered his shot against the bar when under pressure from a defender.
Begovic was called upon six minutes later when Leighton Baines found Nikica Jelavic unmarked inside the box, but he could only direct his header straight at the Stoke keeper when he should have done better.
There was nothing Begovic could do about Everton's opener, however, as Mirallas calmly slotted the ball past him to finish off a fine solo effort having run with it from the edge of his own penalty area.
The hosts almost doubled their lead in the 38th minute when Jelavic powered a free kick towards goal, but Begovic scurried across his goal to keep the ball out.
Mirallas came close to getting another just before the break when he found space to shoot on the edge of the area, but his dragged shot ended up wide of the mark.