We consider the approximation of adjoint-based derivatives for discontinuous solutions of the Cauchy problem associated to one-dimensional nonlinear non-conservative hyperbolic systems. We first derive the adjoint equations in strong form with a discontinuous primal solution together with the associated jump relations across the discontinuity. The adjoint solution may be discontinuous at the discontinuity in contrast to the case of conservative systems. Then, we consider first-order finite volume (FV) approximations to the primal problem and show that, using the Volpert path family of schemes, the discrete adjoint solution is consistent with the strong form adjoint solution. Numerical experiments are shown for a nonlinear 2 × 2 system with a genuinely nonlinear (GNL) field and a linearly degenerate (LD) field associated to the non-conservative product.