An early two-goal salvo from Robert Glatzel and Junior Hoilett earned Cardiff victory as the Owls gave themselves just too much to do at Hillsborough.
Tom Lees put them on the right course with his second strike in as many games but the mountain was too high against a City side largely content to sit on their advantage.
Prior to going 2-0 down, the Owls had actually looked the more likely as they began on the front foot.
But the visitors hit a quickfire double to assume immediate control when first, on five minutes, Glatzel pounced on a loose ball 12 yards out and calmly found the bottom corner.
Then only three minutes later Lee Tomlin darted down the right flank and looked up to find the advancing Hoilett, who made no mistake with a low arrowed drive.
But the shellshocked Owls responded well and twice came close to halving the arrears. Sam Winnall had a snapshot blocked by Jazz Richards before Neil Etheridge superbly diverted a Barry Bannan curler over the crossbar.
And on 18 minutes it was 2-1 as Wednesday were deservedly back in the game. Bannan swung a deep corner to the back post, where Lees climbed high to head emphatically past Etheridge.
The goal sparked the hosts forward and they bossed possession for the remainder of the first half.
With a little fortune, they may have gone down the tunnel on level terms too, as an almighty goalmouth scramble somehow saw Cardiff clear their lines after Dominic Iorfa and Lees each went close.
The second half started as the first had ended, with Wednesday in the ascendancy and the Bluebirds, with a rigid backline, out to frustrate.
Adam Reach almost unlocked the door with a dipping shot from distance and Etheridge dived to deny Jacob Murphy.
The Bluebirds were limited to counter attacks and one such move saw Marlon Pack drill inches past the post.
It was all Wednesday in the closing stages but the home side could not find a way through and they were left to rue a nightmare start that led to a fruitless finish.
Wednesday: Dawson, Palmer (Börner 63), Iorfa (Rhodes 58), Lees, Fox, Reach, Harris, Bannan, Hutchinson, Murphy (Nuhiu 79), Winnall