It was the first taste of success for Everton's new coach Ronald Koeman, whose side fell behind after Northern Ireland defender McAuley stole in at the far post to head home a Craig Gardner corner.
Everton enjoyed the lion's share of the possession in the opening period and made it count on the stroke of halftime when Mirallas beat goalkeeper Ben Foster with a slick finish after incisive passing by Barry and Ross Barkley.
Everton looked a different side after fit-again striker Romelu Lukaku came on as a first-half substitute and Barry capped a memorable afternoon when he nodded in the winner on the hour after Mason Holgate's touch fell kindly for him.
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(Reporting by Zoran Milosavljevic; Editing by Ed Osmond)
