It is tough to detect unexpected drug–drug interactions (DDIs) in poly-drug treatments because of high costs and clinical limitations. Computational approaches, such as deep learning-based approaches, are promising to screen potential DDIs among numerous drug pairs. Nevertheless, existing approaches neglect the asymmetric roles of two drugs in interaction. Such an asymmetry is crucial to poly-drug treatments since it determines drug priority in co-prescription. This paper designs a directed graph attention network (DGAT-DDI) to predict asymmetric DDIs. First, its encoder learns the embeddings of the source role, the target role and the self-roles of a drug. The source role embedding represents how a drug influences other drugs in DDIs. In contrast, the target role embedding represents how it is influenced by others. The self-role embedding encodes its chemical structure in a role-specific manner. Besides, two role-specific items, aggressiveness and impressionability, capture how the number of interaction partners of a drug affects its interaction tendency. Furthermore, the predictor of DGAT-DDI discriminates direction-specific interactions by the combination between two proximities and the above two role-specific items. The proximities measure the similarity between source/target embeddings and self-role embeddings. In the designated experiments, the comparison with state-of-the-art deep learning models demonstrates the superiority of DGAT-DDI across a direction-specific predicting task and a direction-blinded predicting task. An ablation study reveals how well each component of DGAT-DDI contributes to its ability. Moreover, a case study of finding novel DDIs confirms its practical ability, where 7 out of the top 10 candidates are validated in DrugBank.