Summary: We propose a method to modify a polygonal mesh in order to fit the zero-isoline of a level set function by extending a standard body-fitted strategy to a tessellation with arbitrarily-shaped elements. The novel level set-fitted approach, in combination with a Discontinuous Galerkin finite element approximation, provides an ideal setting to model physical problems characterized by embedded or evolving complex geometries, since it allows us skipping any mesh post-processing in terms of grid quality. The proposed methodology is firstly assessed on the linear elasticity equation, by verifying the approximation capability of the level set-fitted approach when dealing with configurations with heterogeneous material properties. Successively, we combine the level set-fitted methodology with a minimum compliance topology optimization technique, in order to deliver optimized layouts exhibiting crisp boundaries and reliable mechanical performances. An extensive numerical test campaign confirms the effectiveness of the proposed method.