We explore a novel application of zero-dimensional persistent homology from Topological Data Analysis (TDA) for bracketing zero-crossings of both one-dimensional continuous functions, and uniformly sampled time series. We present an algorithm and show its robustness in the presence of noise for a range of sampling frequencies. In comparison to state-of-the-art software-based methods for finding zeros of a time series, our method generally converges faster, provides higher accuracy, and is capable of finding all the roots in a given interval instead of converging only to one of them. We also present and compare options for automatically setting the persistence threshold parameter that influences the accurate bracketing of the roots.