This paper explores a new control strategy to suppress the libration of a partial space elevator by applying torque control at the climber only. The control torque at the climber is generated by reaction wheels. A new dynamic model of the partial space elevator is developed by integrating the attitude dynamics of the climber into the classical two-piece dumbbell model. The influence of the climber attitude motion on the libration motion of the partial space elevator is analyzed with and without the control torque at the climber. Based on the characteristics of the system dynamics, a novel reaction wheel control strategy is proposed to suppress both the libration of the elevator and the climber attitude motion. Such a new control strategy is implemented by an optimal control scheme. The simulation results validate the effectiveness of the proposed control strategy, which provides a new alternative method to stabilize the libration of the partial space elevator for fast transfer of cargo with constant speeds. To avoid the adverse effect on the reaction wheel due to the frequent switch of rotation direction, a control strategy by two unidirectional reaction wheels is proposed to achieve the same control effect, where the reaction wheels rotate in only one direction.