The sensitivity equation method aims at estimating the derivative of the solution of partial differential equations with respect to a parameter of interest. The objective of this work is to investigate the ability of an isogeometric Discontinuous Galerkin (DG) method to evaluate accurately sensitivities with respect to shape parameters originating from Computer-Aided Design (CAD), in the context of compressible aerodynamics. The isogeometric DG method relies on Non- Uniform Rational B-Spline representations, which allow to define a high-order numerical scheme for Euler/Navier-stokes equations, fully consistent with CAD geometries. We detail how this formulation can be exploited to construct an efficient and accurate approach to evaluate shape sensitivities. A particular attention is paid to the treatment of boundary conditions for sensitivities, which are more tedious in the case of geometrical parameters. The proposed methodology is first verified on a test- case with analytical solution and then applied to two more demanding problems, that concern the inviscid flow around an airfoil with its camber as shape parameter and the unsteady viscous flow around a three-element airfoil with the positions of slat and flap as parameters.