Barry put a superior Everton ahead with a long-range strike after 23 minutes at Goodison Park before Mirallas doubled the lead from a free-kick just before the hour.
It was only in the latter stages that Norwich finally started to trouble the home defence but Robert Snodgrass' header against the post was the closest they came.
Everton boss Roberto Martinez may have concerns about how his side's concentration wandered towards the end but victory wrapped up a fine day for the Toffees.
It began with them completing the signing of former Celtic winger Aiden McGeady from Spartak Moscow and ended with them climbing back above Liverpool into the Champions League positions.
McGeady was paraded on the pitch prior to kick-off and the positive atmosphere on a day already dedicated to junior fans put a spring in Everton's step.
Among the initiatives for children were a separate edition of the programme and youngsters shadowing club staff, including the stadium announcer and - before the action began - manager Roberto Martinez.
On the field, Romelu Lukaku drove forward early on and shot wide before glancing a header the wrong side of the post.
Tim Howard came out to deny Gary Hooper as Norwich struggled to make inroads and Mirallas saw a long-range effort deflected over.
Leighton Baines also missed the target from distance but there was no denying Barry as he struck from 20 yards.
The on-loan Manchester City midfielder was given space to take aim as he pushed forward and he sent a powerful shot swerving into the top corner beyond John Ruddy.
Norwich almost responded quickly as Ricky van Wolfswinkel, making his first league start since October, raced through and combined with Snodgrass.
He found space to shoot from distance but Howard produced a fine save to deny the Dutchman a first goal since the reverse fixture between the same clubs on the opening day of the season.
It proved a rare attack for the Canaries as Everton lifted the tempo, with Leon Osman earning applause from Martinez as he chased back to win possession.
That sparked another break but Seamus Coleman was unable to drive home his sixth goal in 10 games when the ball made it way to him inside the box.
Barry got into shooting range again but this time the ball was fed to Osman, whose effort was deflected behind.
Everton perhaps lost a little focus in the latter stages of the first half but Norwich were unable to make the most of some possession and set-pieces.
The pace eased off a the start of second period but Everton's superiority was not diminished and it seemed only a matter of time before they increased their lead.
James McCarthy took aim from outside the area after being teed up by Mirallas but it was deflected for another corner.
Norwich did muster a chance for Van Wolfswinkel but he blazed well over.
The second goal arrived just before the hour after a driving run by Baines was crudely halted by Ryan Bennett on the edge of the area.
Mirallas was given the honours and curled a fine shot over the wall that beat Ruddy at his near post.
Lukaku thought he had added a third soon after sweeping past Ruddy and bundling the ball in from a tight angle but the flag had long been raised for offside.
The Canaries struggled to gain any meaningful possession but substitute Nathan Redmond gave them a flash of hope when he beat Coleman on the left and forced Howard to save with his feet.
Encouraged, Bradley Johnson then drove forward to lash one into the side-netting.
Everton's forgotten man John Heitinga, who recently snubbed a move to West Ham, came off the bench as Martinez attempted to tighten up at the back.
Yet Norwich appeared to be growing in confidence and Snodgrass saw a header rebound off the post from a Redmond cross.
It proved the last meaningful opportunity as Everton dug in to secure three points.