The Blues hit back through James McFadden late on but Owen Coyle's men held on for a deserved victory to claim 14th spot.
Lee Bowyer wasted a great chance to give the visitors a second-minute lead when he dragged his shot horribly wide.
Joe Hart then rushed from his line to save with his feet from an unmarked Matt Taylor after Jack Wilshere had picked him out with a wonderful pass.
Chances continued to flow with Zat Knight's 25-yard curler clipping the outside of the post before Cameron Jerome wasted two good opportunities to put his side in front.
And Davies took advantage of Jerome's profligacy when he headed his ninth goal of the season from Paul Robinson's free-kick on 33 minutes.
Birmingham thought they were level six minutes later when McFadden had the ball in the back of the net but his effort was ruled out for handball. After Jussi Jaaskelainen had saved his first shot the Scotland international scooped in the rebound with his upper arm.
The Bolton goalkeeper was back in the action at the start of the second half when he pushed away Roger Johnson's low shot through a crowded penalty area.
And it was the Trotters that doubled their advantage on the hour mark. On-loan Manchester City winger Vladimir Weiss surged into the area before teeing up Klasnic who superbly slotted home.
Birmingham were handed a lifeline with 14 minutes left when the referee pointed to the spot for a non-existent push by Robinson on Christian Benitez.
The Blues substitute saw his weak spot-kick saved by Jaaskelaien, who then denied Bowyer before McFadden bundled the rebound home after a mad scramble on the goal-line.