Graph Neural Networks (GNNs) have demonstrated superior performance in node classification tasks across diverse applications. However, their vulnerability to adversarial attacks, where minor perturbations can mislead model predictions, poses significant challenges. This study introduces GOttack, a novel adversarial attack framework that exploits the topological structure of graphs to undermine the integrity of GNN predictions systematically. By defining a topology-aware method to manipulate graph orbits, our approach generates adversarial modifications that are both subtle and effective, posing a severe test to the robustness of GNNs. We evaluate the efficacy of GOttack across multiple prominent GNN architectures using standard benchmark datasets. Our results show that GOttack outperforms existing state-of-the-art adversarial techniques and completes training in approximately 55% of the time required by the fastest competing model, achieving the highest average misclassification rate in 155 tasks. This work not only sheds light on the susceptibility of GNNs to structured adversarial attacks but also shows that certain topological patterns may play a significant role in the underlying robustness of the GNNs. Our Python implementation is shared at https://github.com/cakcora/GOttack.