This paper proposes an output feedback control method for hydraulic servo systems with unknown dynamics (e.g., actuator dynamics and load variations). This control is an approximation-free approach, and does not need unmeasurable system states. To avoid using tedious backstepping scheme, we introduce a coordinate transform to reformulate the high-order hydraulic servo system model in the strict-feedback form into a Brunovsky form. Then, the high-order sliding mode differentiator(HSMD) is used to reconstruct the unknown states of the derived Brunovsky model. Moreover, a prescribed performance function(PPF) characterizing the error convergence rate, maximum overshoot and steady-state error is utilized in the control design to guarantee both the transient and steady-state performance. To accommodate the lumped unknown dynamics, an unknown system dynamic estimator(USDE) with only one tuning parameter is developed, which retains exponential error convergence and thus helps to achieve precise position tracking control. Simulation results verify the effectiveness of the presented method.