Roger Johnson equalised a Leon Clarke strike on the stroke of half-time, but despite an improved second-half display, it was former Bluebird Deon Burton who struck the winner for the Owls.
The Bluebirds welcomed back skipper Darren Purse and there were also starts for David Forde and Warren Feeney.
Michael Chopra almost worked his way through midway in the half from a Peter Whittingham cross but Iain Turner came out quickly to block his shot.
A stinging drive from Joe Ledley struck the post with Turner beaten as City tried to up the pressure late in the half, but it was the Owls who made the breakthrough in the 39th minute.
Purse was dragged out of position and Clarke found himself on his own in front of goal to collect Burton's cross and, although there was a suspicion of handball, he slotted home from close range.
On the stroke of half-time, Johnson rose high to head in a deep cross from Kevin McNaughton.
Two minutes after the break Ledley's cross was headed goalwards by Feeney but Turner stretched to palm it away and then Chopra tried an overhead kick that rolled wide.
Wednesday regained the lead when Forde failed to hold on as Burton headed a Chris Brunt shot in the 66th minute. Purse made an attempt to clear but Burton followed up to stab home from close range.
Steven Thompson and Matthew Green were brought on as City tried to rescue a point, but the visiting defence held out although Green did hit a great chance into the side-netting in the closing minutes.
In the last minute Simon Walton was sent off for the third time this season for pole-axing Peter Gilbert just before Johnson had a header cleared off the line by Lee Bullen.