Carlos Mendes Gomes sent Fleetwood into the fifth round of the FA Cup for the first time after a deserved 1-0 win in this all League One tie.
The on-loan Luton winger climbed highest to meet Phoenix Patterson’s 60th-minute free-kick to score his eighth goal of the campaign.
Victory for Scott Brown’s side earned them a last-16 showdown with Sky Bet Championship leaders Burnley.
But having inflicted Wednesday’s first defeat in 15 games in all competitions and knocked out QPR in round three, the Clarets will hold no fears for the Cod Army.
Fleetwood took full advantage of an Owls team that contained 10 changes from Saturday’s table-topping win over Plymouth.
Promise Omochere struck a low shot against a post in the first half and missed further chances, all from a supply line provided by Patterson.
Wednesday were second best for most of the night. When they did fashion chances, Fisayo Dele-Bashiri’s shot was easily saved by Jay Lynch, while Lee Gregory headed against the crossbar in the opening period.
