We present a numerically efficient steady-state finite-element solver for electromechanical devices incorporating magnetic saturation in which the magnetization of ferromagnetic materials is modeled as a field-dependent equivalent current density. We use a Lagrangian representation of continuum variables, thereby removing numerical instabilities due to the Peclet effect but precluding the use of standard harmonic balance methods. The shooting-Newton method is therefore used to calculate the steady-state behavior. A matrix-free Krylov-subspace linear solver, the generalized minimum residuals method (GMRES), dramatically reduces the computational burden by eliminating the need to calculate the shooting-Newton Jacobian. Simulation results from a synchronous reluctance motor model with 10 288 nodes and 4935 elements confirm that the proposed method requires much less computation time than running transient analysis until convergence.