Fairness-aware graph learning has become increasingly important amid growing concerns about algorithmic bias in networked data. Among existing approaches, counterfactual fairness is particularly appealing as it seeks to eliminate unfairness at its causal origin by ensuring that predictions remain invariant in counterfactual worlds where sensitive attributes are altered. However, most existing methods assume that all observed variables are directly influenced by sensitive attributes, an overly strong and often unrealistic assumption in real-world graphs. To address this limitation, we propose Graph Counterfactual Fairness (GCFair), a novel framework that achieves counterfactual fairness by explicitly identifying and disentangling the subsets of node features and graph structures genuinely affected by sensitive attributes. This principled joint disentanglement enables the generation of authentic counterfactual instances that selectively modify only sensitive-related information while preserving all sensitive-irrelevant factors. Extensive experiments show that GCFair effectively mitigates bias and outperforms state-of-the-art fairness methods in both counterfactual fairness and predictive accuracy.