Two goals from striker Andrew Johnson earned Fulham their first away win of the season and stretched their unbeaten run to 10 matches.
Johnson gave the Premier League side the lead with a 12th minute goal but the Owls drew level on 20 minutes thanks to a superb long-range effort from defender Tommy Spurr.
The Championship club, who went into the tie with no fewer than 10 players on the injured list, had opportunities to take the lead, with the fit-again Jermain Johnson twice squandering clear-cut chances before Johnson grabbed the winner three minutes from the end.
