Maik Taylor had to be alert in the fourth minute as a cross from Peter Whittingham was met close in by Paul Parry, but his header went straight at the keeper.
One minute later Birmingham went ahead with a simple strike by James McFadden. Quincy Owusu-Abeyie cut inside before slipping a pass to McFadden, who calmly slotted the ball into the net with Darren Purse appealing for offside.
Ledley fired in an 18th minute shot which took a deflection as the home side pinned the Blues back in defence.
Ross McCormack then beat Taylor to a long ball through the middle but took it too wide to give the visitors' defence chance to get back in numbers.
In the 34th minute Birmingham skipper Lee Carsley cleared a Roger Johnson header off the line with Taylor beaten.
Four minutes before the interval and totally against the run of play Owusu-Abeyie netted Birmingham's second, when he received from a throw-in and fired into the corner of the net with the home defence motionless.
Neither side made changes at the interval and it was still Cardiff who were making the running, with Johnson and McCormack both going close, while Birmingham seemed content to soak up the pressure.
Johnson went close with a header from a Whittingham corner, but Radhi Jaidi and Liam Ridgewell were more than a match for Cardiff's strikers.
A neat pass from Kevin McNaughton gave McCormack a sight on goal in the 78th minute, but his shot struck the top of the crossbar.
The busy McCormack reduced the deficit in the 87th minute when he flashed a low shot into the net and although the Bluebirds pushed for an equaliser, Birmingham held out for the victory.