Power system state estimation (PSSE) is a critical task for grid operation efficiency and system stability. Physical laws dictate quadratic relationships between observable quantities and voltage state variables, hence rendering the PSSE problem nonconvex and NP-hard. Existing SE solvers largely rely on iterative optimization methods or semidefinite relaxation (SDR) techniques. Even when based on noiseless measurements, convergence of the former is sensitive to the initialization, while the latter is challenged by small-size measurements especially when voltage magnitudes are not available at all buses. At the price of running time, this paper proposes a novel feasible point pursuit (FPP)-based SE solver, which iteratively seeks feasible solutions for a nonconvex quadratically constrained quadratic programming reformulation of the weighted least-squares (WLS) SE problem. Numerical tests corroborate that the novel FPP-based SE markedly improves upon the Gauss-Newton based WLS and SDR-based SE alternatives, also when noisy measurements are available.