Subject to a voltage, dielectric elastomers (DEs) undergo large deformations, resulting in a wide array of applications in actuators and energy generators. In this paper, a novel potential application of DEs is presented that employs a DE as a vibration damper to achieve the vibration attenuation of a spring oscillator. Based on the developed model, the passive vibration control of the spring oscillator is investigated by utilizing the viscoelastic damping of the DE itself. Subsequently, the active vibration attenuation is studied by applying alternating oppositely-phased voltages at different frequencies, which applies to DEs with a relatively lower viscoelastic damping coefficient. In order to obtain a tunable and controllable vibration attenuation process, the corresponding desired voltage is calculated through the prescribed trend of stretch. The results indicate that the vibration attenuation can be tuned and specified by applying the calculated voltage.