Graph Neural Networks (GNNs) have demonstrated their powerful capability in learning representations for graph-structured data. Consequently, they have enhanced the performance of many graph-related tasks such as node classification and graph classification. However, it is evident from recent studies that GNNs are vulnerable to adversarial attacks. Their performance can be largely impaired by deliberately adding carefully created unnoticeable perturbations to the graph. Existing attacking methods often produce perturbation by adding/deleting a few edges, which might be noticeable even when the number of modified edges is small. In this paper, we propose a graph rewiring operation to perform the attack. It can affect the graph in a less noticeable way compared to existing operations such as adding/deleting edges. We then utilize deep reinforcement learning to learn the strategy to effectively perform the rewiring operations. Experiments on real-world graphs demonstrate the effectiveness of the proposed framework. To understand the proposed framework, we further analyze how its generated perturbation impacts the target model and the advantages of the rewiring operations. The implementation of the proposed framework is available at https://github.com/alge24/ReWatt.