Problems in composite materials, multiphase fluids, fluid-structure interaction and other fields are modelled imposing different PDEs on different spatial regions of a common computational domain and enforcing coupling conditions at the interfaces. For the numerical treatment of such problems a new class of discontinuous Galerkin schemes is introduced. The schemes are derived from a relaxation approach and do not require information on the eigenstructure of the coupled systems, which simplifies the computation of suitable coupling data and thus provides a black-box-like approach for the user. To preserve the coupling-condition at the interface Jin-Xin relaxation is used, for which a modification of the condition is required. A simple algorithm guides this construction for general coupled hyperbolic problems. The schemes are endowed with higher order time discretization by means of SSP-Runge-Kutta methods derived from an asymptotic preserving implicit-explicit treatment of the relaxation system. The preservation of simple coupling rules, such as Kirchhoff conditions, by the scheme on the discrete level is shown. In a case study the application to multi-dimensional fluid-structure coupling problems employing the compressible Euler equations and a linear elastic model is discussed.