Bluebirds boss Dave Jones opted to keep Joe Ledley as his captain with Stephen McPhail left on the bench while Michael Chopra passed a late fitness test and started in attack.
Jay Bothroyd had the first scoring opportunity in the eighth minute when he turned quickly to fire Roger Johnson's downward header over the bar from close range.
Wednesday hit back and Peter Enckelman had to dive at the feet of Luke Boden and then Marcus Tudgay as the Owls threatened to open the scoring.
Paul Parry then sent Chopra racing down the flank in the 13th minute and, when Paul Parry crossed, Bothroyd was wide with a snap header.
Midway through the half another Parry cross was headed by Johnson but Owls keeper Lee Grant moved to his left to safely gather.
Gavin Rae was brought down on the edge of the area in the 25th minute after Chopra had slipped him through but Chopra's free-kick was charged down.
The first half drew to a close with City on top but they were unable to break down a stubborn Wednesday back line.
Five minutes into the second half that all changed when City went ahead. Wayne Routledge crossed the ball into the middle, Bothroyd headed down and Johnson struck a first-time shot that took a deflection as it headed for the left-hand corner of the Owls net.
City upped the tempo and Routledge had Grant palming away a shot after the impressive Bothroyd had freed the winger. Bothroyd then fired in a superb low shot on the hour that forced Grant to dive low to gather.
Five minutes later Routledge set up Ledley who raced into the box where he was brought down in full flight. Chopra took the spot-kick and rammed it into the net to double City's lead.
In the 69th minute Bothroyd smashed a shot against the post and Chopra slotted home the rebound but the flag had been raised for offside.
The Bluebirds were well on top at this stage but Wednesday managed to keep them out for the rest of the match.