Cross-domain graph anomaly detection (CD-GAD) describes the problem of detecting anomalous nodes in an unlabelled target graph using auxiliary, related source graphs with labelled anomalous and normal nodes. Although it presents a promising approach to address the notoriously high false positive issue in anomaly detection, little work has been done in this line of research. There are numerous domain adaptation methods in the literature, but it is difficult to adapt them for GAD due to the unknown distributions of the anomalies and the complex node relations embedded in graph data. To this end, we introduce a novel domain adaptation approach, namely Anomaly-aware Contrastive alignmenT (ACT), for GAD. ACT is designed to jointly optimise: (i) unsupervised contrastive learning of normal representations of nodes in the target graph, and (ii) anomaly-aware one-class alignment that aligns these contrastive node representations and the representations of labelled normal nodes in the source graph, while enforcing significant deviation of the representations of the normal nodes from the labelled anomalous nodes in the source graph. In doing so, ACT effectively transfers anomaly-informed knowledge from the source graph to learn the complex node relations of the normal class for GAD on the target graph without any specification of the anomaly distributions. Extensive experiments on eight CD-GAD settings demonstrate that our approach ACT achieves substantially improved detection performance over 10 state-of-the-art GAD methods. Code is available at https://github.com/QZ-WANG/ACT.