Clusters of viral pneumonia occurrences over a short period may be a harbinger of an outbreak or pandemic. Rapid and accurate detection of viral pneumonia using chest X-rays can be of significant value for large-scale screening and epidemic prevention, particularly when other more sophisticated imaging modalities are not readily accessible. However, the emergence of novel mutated viruses causes a substantial dataset shift, which can greatly limit the performance of classification-based approaches. In this paper, we formulate the task of differentiating viral pneumonia from non-viral pneumonia and healthy controls into a one-class classification-based anomaly detection problem. We therefore propose the confidence-aware anomaly detection (CAAD) model, which consists of a shared feature extractor, an anomaly detection module, and a confidence prediction module. If the anomaly score produced by the anomaly detection module is large enough, or the confidence score estimated by the confidence prediction module is small enough, the input will be accepted as an anomaly case (i.e., viral pneumonia). The major advantage of our approach over binary classification is that we avoid modeling individual viral pneumonia classes explicitly and treat all known viral pneumonia cases as anomalies to improve the one-class model. The proposed model outperforms binary classification models on the clinical X-VIRAL dataset that contains 5,977 viral pneumonia (no COVID-19) cases, 37,393 non-viral pneumonia or healthy cases. Moreover, when directly testing on the X-COVID dataset that contains 106 COVID-19 cases and 107 normal controls without any fine-tuning, our model achieves an AUC of 83.61% and sensitivity of 71.70%, which is comparable to the performance of radiologists reported in the literature.