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We introduce a new method to estimate the probability that an extragalactic transient source is associated with a candidate host galaxy. This approach relies solely on simple observables: sky coordinates and their uncertainties, galaxy fluxes, and angular sizes. The formalism invokes Bayes' rule to calculate the posterior probability from the galaxy prior P(O), observables x, and an assumed model for the true distribution of transients in/around their host galaxies. Using simulated transients placed in the well-studied Cosmic Evolution Survey field, we consider several agnostic and physically motivated priors and offset distributions to explore the method sensitivity. We then apply the methodology to the set of 13 fast radio bursts (FRBs) localized with an uncertainty of several arcseconds. Our methodology finds nine of these are securely associated to a single host galaxy,. We examine the observed and intrinsic properties of these secure FRB hosts, recovering distributions similar to those found in previous works. Furthermore, we find a strong correlation between the apparent magnitude of the securely identified host galaxies and the estimated cosmic dispersion measures of the corresponding FRBs, which results from the Macquart relation. Future work with FRBs will leverage this relation and other measures from the secure hosts as priors for future associations. The methodology is generic to transient type, localization error, and image quality. We encourage its application to other transients where host galaxy associations are critical to the science, e.g., gravitational wave events, gamma-ray bursts, and supernovae. We have encoded the technique in Python on GitHub: https://github.com/FRBs/astropath. [ABSTRACT FROM AUTHOR]