In this work, we present a numerical method for the resolution of transmission problems with non-conformal meshes which preserves the optimal rates of convergence in space. The smooth extension method is a fictitious domain approach based on a control formulation stated as a minimization problem, that we prove to be equivalent to the initial transmission problem. Formulated as a minimization problem, the transmission problem can be solved with standard finite element function spaces and usual optimization algorithms. The method is applied to different transmission problems (Laplace, Stokes and a fluid-structure interaction problem) and compared to standard finite element methods.