Even for simple diffusion processes, treating first-passage problems analytically proves intractable for generic barriers and existing numerical methods are inaccurate and computationally costly. Here, we present a novel numerical method that is faster and has more tightly controlled accuracy. Our algorithm is a probabilistic variant of dichotomic search for the computation of first passage times through non-negative homogeneously Hölder continuous boundaries by Gauss-Markov processes. These include the Ornstein-Uhlenbeck process underlying the ubiquitous “leaky integrate-and-fire” model of neuronal excitation. Our method evaluates discrete points in a sample path exactly, and refines this representation recursively only in regions where a passage is rigorously estimated to be probable (e.g. when close to the boundary).