Electromagnetic (EM) scattering by particles in an absorbing host medium is frequently encountered in practical applications, which makes the conventional EM scattering theory controversial and most of the theoretical methods for EM scattering inapplicable. Most of the relevant works in literature are confined to spherical particles. In this work, we develop the discrete dipole approximation (DDA) method for EM scattering by an arbitrary particle immersed in an absorbing host medium. We elaborate how the near- and far-field scattering quantities can be calculated by DDA. The accuracy of DDA is validated by comparison with the apparent and inherent scattering quantities of spherical particles computed by exact Mie theory. Then EM extinction by non-absorbing spheroids in absorbing host medium is studied by DDA. We find that particles that are prolonged in the incident direction are more likely to produce a negative apparent extinction, which is also supported by the near-field electric field distribution. The DDA method we develop will be useful and flexible in the study of EM scattering by particles in absorbing host medium.