In this paper, we study the saturation control problem for linear time-invariant (LTI) systems subject to asymmetric actuator saturation under a switching control framework. The LTI plant with asymmetric saturation is first transformed to an equivalent switched linear model with each subsystem subject to symmetric actuator saturation, based on which a dwell-time switching controller augmented with a controller state reset is then developed by using multiple Lyapunov functions. The controller synthesis conditions are formulated as linear matrix inequalities (LMIs), which can be solved efficiently. Simulation results are also included to illustrate the effectiveness and advantages of the proposed approach.