Compressible fluid flows are currently mostly simulated using shock-capturing Finite Volume methods or Discontinuous Galerkin (DG) methods, in which the quest for higher-order convergence rates entails numerical oscillations near shocks. A classical approach to prevent these oscillations is to add artificial viscosity or slope-limiting, smearing out at the same time the physical acoustic waves. It has been recently proposed to revisit shock-fitting methods in the context of cut-cell methods or optimization-based mesh moving methods in order to circumvent these issues. These methods have the advantage that the solution becomes piecewise-continuous with an accurate tracking of the discontinuity surface using the Rankine-Hugoniot conditions. In the present work, we propose to extend the cut-cell method presented in [Puscas et al. IJNME 2015] for fluid-structure interaction to handle shock-fitting in a cut-cell DG framework. The main idea is to modify the expression of the additional boundary flux in the formulation and relating it to the normal velocity of the discontinuity surface through the resolution of a one-dimensional Riemann problem. Numerical results will demonstrate the improved convergence of the method and discuss possible advantages and drawbacks compared to shock-capturing methods and other shock-fitting methods.