This paper considers an important Graph Anomaly Detection (GAD) task, namely open-set GAD, which aims to train a detection model using a small number of normal and anomaly nodes (referred to as seen anomalies) to detect both seen anomalies and unseen anomalies (i.e., anomalies that cannot be illustrated the training anomalies). Those labelled training data provide crucial prior knowledge about abnormalities for GAD models, enabling substantially reduced detection errors. However, current supervised GAD methods tend to over-emphasise fitting the seen anomalies, leading to many errors of detecting the unseen anomalies as normal nodes. Further, existing open-set AD models were introduced to handle Euclidean data, failing to effectively capture discriminative features from graph structure and node attributes for GAD. In this work, we propose a novel open-set GAD approach, namely normal structure regularisation (NSReg), to achieve generalised detection ability to unseen anomalies, while maintaining its effectiveness on detecting seen anomalies. The key idea in NSReg is to introduce a regularisation term that enforces the learning of compact, semantically-rich representations of normal nodes based on their structural relations to other nodes. When being optimised with supervised anomaly detection losses, the regularisation term helps incorporate strong normality into the modelling, and thus, it effectively avoids over-fitting the seen anomalies and learns a better normality decision boundary, largely reducing the false negatives of detecting unseen anomalies as normal. Extensive empirical results on seven real-world datasets show that NSReg significantly outperforms state-of-the-art competing methods by at least 14% AUC-ROC on the unseen anomaly classes and by 10% AUC-ROC on all anomaly classes.