This technical note addresses the robust finite-horizon output feedback control problem for a class of uncertain discrete stochastic nonlinear time-varying systems with both sensor and actuator saturations. In the system under investigation, all the system parameters are allowed to be time-varying, the parameter uncertainties are assumed to be of the polytopic type, and the stochastic nonlinearities are described by statistical means which can cover several classes of well-studied nonlinearities. The purpose of the problem addressed is to design an output feedback controller, over a given finite-horizon, such that the disturbance attenuation level is guaranteed for the nonlinear stochastic polytopic system in the presence of saturated sensor and actuator outputs. Sufficient conditions are first established for the robust performance through intensive stochastic analysis, and then a recursive linear matrix inequality (RLMI) approach is employed to design the desired output feedback controller achieving the prescribed disturbance rejection level. Simulation results demonstrate the effectiveness of the developed controller design scheme.