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The first chance of the match fell the way of the visitors as Clint Hill latched on to a quick free kick, only to nod his header just wide.
The visitors found themselves with a mountain to climb just two minutes later, however, as Dunne conceded a penalty and was shown a straight red card for a foul on Best inside the area.
Maguire was the man entrusted with the penalty duties and he made no mistake, sending Brian Murphy the wrong way to give his side the lead.
Murphy was forced into action again right on the stroke of half time when he made a fine stop to deny Atdhe Nuhiu in the 45th minute.
The Owls did eventually double their lead six minutes after the restart when Oguchi Onyewu picked out Best inside the box, allowing the striker to turn and fire an effort into the bottom corner.
Things got even better for the hosts with 19 minutes remaining as they extended their lead to three goals, Lewis Buxton finding the back of the net with a fine effort.
Wednesday continued to look for a fourth and came close to getting it with 10 minutes remaining as Lavery cut inside before drawing a good save from Murphy.
Jermaine Johnson also went close twice late on for the hosts, but they couldn't add to their lead as they settled for a convincing 3-0 victory.