We investigated both classical and quantum properties of a damped harmonic oscillator with a time-variable elastic coefficient using invariant operator method. We acquired the energy eigenvalues, uncertainties and probability densities for several types of wave packet. The probability density corresponding to the displaced minimum wave packet expressed in terms of the time-dependent Gaussian function. The displaced minimum wave packet not only be attenuated but also oscillates about x = 0. We confirmed that there exist correspondence between quantum and classical behaviors for the time-dependent damped harmonic oscillator.