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First-passage times provide invaluable insight into fundamental properties of stochastic processes. Yet, various forms of gating mask first-passage times and differentiate them from actual detection times. For instance, imperfect conditions may intermittently gate our ability to observe a system of interest, such that exact first-passage instances might be missed. In other cases, e.g., certain chemical reactions, direct observation of the molecules involved is virtually impossible, but the reaction event itself can be detected. However, this instance need not coincide with the first collision time since some molecular encounters are infertile and hence gated. Motivated by the challenge posed by such real-life situations we develop a universal -- model-free -- framework for the inference of first-passage times from the detection times of gated first-passage processes. In addition, when the underlying laws of motions are known, our framework also provides a way to infer physically meaningful parameters, e.g. diffusion coefficients. Finally, we show how to infer the gating rates themselves via the hitherto overlooked short-time regime of the measured detection times. The robustness of our approach and its insensitivity to underlying details are illustrated in several settings of physical relevance.