In this article, we consider a fluid-structure interaction system where the fluid is viscous and compressible and where the structure is a part of the boundary of the fluid domain and is deformable. The fluid is governed by the barotropic compressible Navier-Stokes system whereas the structure displacement is described by a wave equation. We show that the corresponding coupled system admits a unique strong solution for an initial fluid density and an initial fluid velocity in $H^3$ and for an initial deformation and an initial deformation velocity in $H^4$ and $H^3$ respectively. The reference configuration for the fluid domain is a rectangular cuboid with the elastic structure being the top face. We use a modified Lagrangian change of variables to transform the moving fluid domain into the rectangular cuboid and then analyze the corresponding linear system coupling a transport equation (for the density), a heat-type equation, and a wave equation. The corresponding results for this linear system and estimations of the coefficients coming from the change of variables allow us to perform a fixed point argument and to prove the existence and uniqueness of strong solutions for the nonlinear system, locally in time.