Finland international striker Mikael Forssell opened the scoring for the vistors on 35 minutes and Emile Heskey scored a second just two minutes before the break.
Heskey then made it 3-0 just 10 minutes into the second half.
Njazi Kuqi scored Birmingham's fourth in injury time.
Derby: Camp, Mills (Nytanga 80), Jackson, Johnson, Konjic, Mills, Bisgaard (Tugay 70), Doyle, Idiakez, Bolder, Smith (Barnes 84), Rasiak.
Birmingham: Maik Taylor, Melchiot, Gray, Cunningham, Upson, Tebily (Kilkenny h/t), Izzet (Hall h/t), Anderton (Martin Taylor 62), Pennant (Painter 80), Forssell (Morrison 62), Heskey (Kuqi 70).