The Owls were pegged back by Fulham late in the game after dominating at Hillsborough on Saturday.
Matt Smith cancelled out an opener from Stevie May, who came off the bench to find the net.
It was a frustrating afternoon for Wednesday, with Stuart Gray’s side controlling possession and the pace of the contest.
Fulham were on the back foot through the first half but threatened on the counter, with Ross McCormack the main threat.
Tom Hoogland was a whisker away from sending the striker clear on 18 minutes but Jeremy Helan used his full throttle to clear the danger.
By and large, though, it was Wednesday knocking on the door and Chris Maguire forced a diving save from Marcus Bettinelli as the hosts pressed.
But Wednesday suffered a double injury blow before the interval.
On 28 minutes, Liam Palmer was stretchered off, closely followed by a limping Will Keane - Marnick Vermijl and May came on at the back and up front respectively.
Then just before the whistle, the Owls created two openings that might have broken the deadlock.
First, Sam Hutchinson cleverly flicked a Helan feed across the six-yard box but Fulham escaped the danger before May miscued a hooked half volley after an inviting knockdown from Atdhe Nuhiu.
But May was not to be denied and nine minutes into the second period, the Scot made the breakthrough.
Lewis McGugan delivered an accurate ball from the right and May was on the spot to glance a fine header beyond Bettinelli.
The goal gave the Owls a visible lift and Vermijl drove forward before unleashing a fierce shot from distance that Bettinelli pushed past the post.
Fulham were given a start too and Hoogland slammed a rasping hit that took Claude Dielna off his feet as the defender bravely blocked.
And with 16 minutes to go, the Londoners snatched an equaliser against the run of play.
Wednesday survived a penalty scare when Ryan Tunnicliffe tumbled beneath a Vermijl challenge but Fulham scored regardless as Smith promptly clipped a deft shot in off the underside of the bar.
Owls: Westwood, Hutchinson, Palmer (Vermijl 29), Nuhiu, Maguire, Lees, Helan, Lee (Maghoma 90), Dielna, McGugan, Keane (May 41)
Fulham: Bettinelli, Hoogland (Woodrow 69), Stafylidis, Hutchinson, Parker, Smith, Turner, Tunnicliffe, Richards, Kavanagh (Fofana 69), McCormack