We study the numerical approximation of a coupled hyperbolic-parabolic system by a family of discontinuous Galerkin (DG) space-time finite element methods. The model is rewritten as a first-order evolutionary problem that is treated by a unified abstract solution theory. For the discretization in space, generalizations of the distribution gradient and divergence operators on broken polynomial spaces are defined. Since their skew-selfadjointness is perturbed by boundary surface integrals, adjustments are introduced such that the skew-selfadjointness of the discrete counterpart of the total system’s first-order differential operator in space is recovered. Well-posedness of the fully discrete problem and error estimates for the DG approximation in space and time are proved.