Agile localization of anomalous events plays a pivotal role in enhancing the overall reliability of the grid and avoiding cascading failures. This is especially of paramount significance in the large-scale grids due to their geographical expansions and the large volume of data generated. This paper proposes a stochastic graphical framework, by leveraging which it aims to localize the anomalies with the minimum amount of data. This framework capitalizes on the strong correlation structures observed among the measurements collected from different buses. The proposed approach, at its core, collects the measurements sequentially and progressively updates its decision about the location of the anomaly. The process resumes until the location of the anomaly can be identified with desired reliability. We provide a general theory for the quickest anomaly localization and also investigate its application for quickest line outage localization. Simulations in the IEEE 118-bus model are provided to establish the gains of the proposed approach.