Existing cross-network node classification methods are mainly proposed for closed-set setting, where the source network and the target network share exactly the same label space. Such a setting is restricted in real-world applications, since the target network might contain additional classes that are not present in the source. In this work, we study a more realistic open-set cross-network node classification (O-CNNC) problem, where the target network contains all the known classes in the source and further contains several target-private classes unseen in the source. Borrowing the concept from open-set domain adaptation, all target-private classes are defined as an additional unknown class. To address the challenging O-CNNC problem, we propose an unknown-excluded adversarial graph domain alignment (UAGA) model with a separate-adapt training strategy. Firstly, UAGA roughly separates known classes from unknown class, by training a graph neural network encoder and a neighborhood-aggregation node classifier in an adversarial framework. Then, unknown-excluded adversarial domain alignment is customized to align only target nodes from known classes with the source, while pushing target nodes from unknown class far away from the source, by assigning positive and negative domain adaptation coefficient to known class nodes and unknown class nodes. Extensive experiments on real-world datasets demonstrate significant outperformance of the proposed UAGA over state-of-the-art methods on O-CNNC.