We study the problem of identifying anomalies in a low-rank matrix observed with sub-exponential noise, motivated by applications in retail and inventory management. State of the art approaches to anomaly detection in low-rank matrices apparently fall short, since they require that non-anomalous entries be observed with vanishingly small noise (which is not the case in our problem, and indeed in many applications). So motivated, we propose a conceptually simple entrywise approach to anomaly detection in low-rank matrices. Our approach accommodates a general class of probabilistic anomaly models. We extend recent work on entrywise error guarantees for matrix completion, establishing such guarantees for sub-exponential matrices, where in addition to missing entries, a fraction of entries are corrupted by (an also unknown) anomaly model. Viewing the anomaly detection as a classification task, to the best of our knowledge, we are the first to achieve the min-max optimal detection rate (up to log factors). Using data from a massive consumer goods retailer, we show that our approach provides significant improvements over incumbent approaches to anomaly detection.