A well known result by O. Kowalski and J. Szenthe says that any homogeneous Riemannian manifold admits a homogeneous geodesic through any point. This was proved by the algebraic method using the reductive decomposition of the Lie algebra of the isometry group. In previous papers by the author, the existence of a homogeneous geodesic in any homogeneous pseudo-Riemannian manifold and also in any homogeneous affine manifold was proved. In this setting, a new method based on affine Killing vector fields was developed. Using this method, it was further proved that any homogeneous Lorentzian manifold of even dimension admits a light-like homogeneous geodesic and any homogeneous Finsler space of odd dimension admits a homogeneous geodesic. In the present paper, the affine method is further refined for Finsler spaces and it is proved that any homogeneous Berwald space or homogeneous reversible Finsler space admits a homogeneous geodesic through any point.