In modern transportation networks, adversaries can manipulate routing algorithms using false data injection attacks, such as simulating heavy traffic with multiple devices running crowdsourced navigation applications, to mislead vehicles toward suboptimal routes and increase congestion. To address these threats, we formulate a strategically zero-sum game between an attacker, who injects such perturbations, and a defender, who detects anomalies based on the observed travel times of network edges. We propose a computational method based on multi-agent reinforcement learning to compute a Nash equilibrium of this game, providing an optimal detection strategy, which ensures that total travel time remains within a worst-case bound, even in the presence of an attack. We present an extensive experimental evaluation that demonstrates the robustness and practical benefits of our approach, providing a powerful framework to improve the resilience of transportation networks against false data injection. In particular, we show that our approach yields approximate equilibrium policies and significantly outperforms baselines for both the attacker and the defender.