In this work we develop a novel time-domain approach for the simultaneous estimation of the damping parameters for a single degree of freedom oscillator with both viscous and coulomb damping. Our approach leverages zero-dimensional sublevel set persistence — a tool from Topological Signal Processing (TSP) — to analyze the ring down vibration of the signal. Sublevel set persistence is used as it alleviates the need for peak selection when analyzing the time-domain of the signal and provides an alternative noise-robust method for visualizing the damping envelope. We are able to successfully estimate the damping parameters using both a direct approach and a function fitting method. We show that the direct approach is only appropriate for low levels of additive noise, but allows for a less computationally demanding estimation of the parameters. Alternatively, the function fitting method provides accurate estimates for significantly higher levels of additive noise. The results are provided through a numerically simulated example with mixed coulomb and viscous damping. We demonstrate the robustness of our method for accurately estimating both damping parameters for various levels of additive noise, a wide range of sampling frequencies, and both high and low levels of damping. This analysis includes providing suggested limitations of the method when applied to real-world signals.