A state-switched device is conceptually capable of instantaneously changing its mass, stiffness, or damping. Such a device will exhibit different dynamical response properties (modes and resonance frequencies) depending on its current state. A state-switched vibration absorber exploits the state-switching concept for the purposes of enhanced vibration suppression. Between each state switch, it is fundamentally a passive vibration absorber, but one which exhibits a different tuning frequency for each possible state. A state-switched vibration absorber therefore has a greater effective bandwidth than a classical passive absorber. This paper considers the role of damping in the state-switching concept for a simple one-degree of freedom system and for a two-degree of freedom system. Certain values of damping in the system improve performance, while other values hinder the performance of the state-switched absorber, as compared to classical absorbers. The predicted performance of the system also depends upon the particular damping model used, such as proportional, viscous, or modal damping. Damping values also affect the frequency of switch events that occur during the response of the system. In general, a state-switched absorber with optimized damping is more effective at vibration suppression as compared to a classical vibration absorber with optimized damping.