The Owls, who have won four matches on the bounce since an opening-game draw with Alfreton Town, claimed their latest triumph in a convincing manner.
Wednesday continued to turn the screw and were eventually rewarded 19 minutes in thanks to a powerful Tom Lees header at the end of a Ross Wallace corner.
Fernando Forestieri helped the Owls to a deserved second shortly before the hour with a smart finish to make it five in five for the summer.
There was also a first goal for substitute Almen Abdi, who made a positive impact after coming on and was in the right place to round off a fine team move.
Abdi also hit the bar with a drilled effort late on, as the League One opposition struggled to provide any real challenge in the end.