In a dissipative quantum system, we report the dynamic mode decomposition (DMD) analysis of damped oscillation signals. We used a reflection-type pump-probe method to observe time-domain signals, including the coupled modes of long-lived longitudinal optical phonons and quickly damped plasmons (LOPC) at various pump powers. The Fourier transformed spectra of the observed damped oscillation signals show broad and asymmetric modes, making it difficult to evaluate their frequencies and damping rates. We then used DMD to analyze the damped oscillation signals by precisely determining their frequencies and damping rates. We successfully identified the LOPC modes. The obtained frequencies and damping rates were shown to depend on the pump power, which implies photoexcited carrier density. We compared the pump-power dependence of the frequencies and damping rates of the LOPC modes with the carrier density dependence of the complex eigen-energies of the coupled modes by using the non-Hermitian phenomenological effective Hamiltonian. Good agreement was obtained between the observed and calculated dependences, demonstrating that DMD is an effective alternative to Fourier analysis which often fails to estimate effective damping rates.