In this article, we devise a simulation tool of the dynamics of a tethered buoy, which is a mooring system consisting of a rigid floating body (buoy) connected by an elastic cable (mooring line) to the bottom of the fluid environment. The accuracy and robustness of the overall solution algorithm are based on an appropriate choice of three main ingredients. The first ingredient is the use of a novel mixed finite element formulation for the mooring line, which allows a robust modeling of the cable even in the limit of an infinite value of the Young modulus. The second ingredient is the use of quaternions in the equations of motion of the floating body, instead of Euler angles, which avoids angle singularities in the presence of large rotations. The third ingredient is the use of an implicit method (Backward Euler) for time discretization, in conjunction with a damped Newton method for the solution of the resulting nonlinear system, which avoids computational inefficiency of explicit time integration schemes in the presence of quasi-inextensible cables and a fine spatial grid size. The physical soundness of the mooring system model and the accuracy and robustness of the proposed numerical procedure are validated in the study of a tethered buoy operating under static and dynamic industrial-like working conditions.