Elastic wave scattering has received ever-increasing attention in military and medical fields due to obtain high-precision solutions. In this paper, an edge-based smoothed finite element method (ES-FEM) combined with the transparent boundary condition (TBC) is proposed to solve the elastic wave scattering problems by a rigid obstacle, which is embedded in an isotropic and homogeneous elastic medium in two dimensions. The elastic wave scattering problems satisfy Helmholtz equations with coupled boundary obtained by Helmholtz decomposition. Firstly, the TBC of elastic wave scattering is constructed by using the analytical solutions of Helmholtz equations, which can truncate the boundary value problems (BVP) in an unbounded domain into the BVP in a bounded domain. Then the formulations of S-FEM with the TBC are derived for Helmholtz equations with coupled boundary. Finally, some numerical examples illustrate that the proposed ES-FEM with the TBC (ES-FEM-TBC) can work effectively and obtain more stable and more accurate solutions than the standard FEM with the TBC (FEM-TBC) for elastic wave scattering.