Isogeometric analysis method has known a growing interest in engineering fields for the last years, thanks to its ability to strengthen the coupling between Computer Aided Design (CAD) and Finite-Element Analysis (FEA). However, this methodological framework is not so well suited to hyperbolic conservation laws, encountered in problems governed by wave propagation phenomena, such as acoustics, electromagnetics or compressible aerodynamics. In this context, additional stabilization terms may be required, whose calibration is tedious. Therefore, we investigate in this work the adaption of the Discontinuous Galerkin (DG) method to handle geometries defined by Non-Uniform Rational B-Spline (NURBS) boundaries. More specifically, we explain how to generate a DG-compliant computational domain from a set of NURBS boundaries, i.e. a set of rational elements allowing discontinuities in the solution at the interfaces, while relying on an exact description of the boundaries geometry. The proposed approach is based on a combination of classical CAD procedures, such as degree elevation and knot insertion. Then, a DG solving procedure is derived, which uses the same basis for the geometry and solution according to the isogeometric paradigm, and includes a more straightforward stabilization through upwind fluxes. The approach is applied to different hyperbolic conservation law problems, ranging from linear advection to compressible Euler equations. Optimal convergence rates of the method are demonstrated, while the critical role of boundary curvature is reported.