Urban traffic congestion is a critical predicament that plagues modern road networks. To alleviate this issue and enhance traffic efficiency, traffic signal control and vehicle routing have proven to be effective measures. In this paper, we propose a joint optimization approach for traffic signal control and vehicle routing in signalized road networks. The objective is to enhance network performance by simultaneously controlling signal timings and route choices using Multi-Agent Deep Reinforcement Learning (MADRL). Signal control agents (SAs) are employed to establish signal timings at intersections, whereas vehicle routing agents (RAs) are responsible for selecting vehicle routes. By establishing relevance between agents and enabling them to share observations and rewards, interaction and cooperation among agents are fostered, which enhances individual training. The Multi-Agent Advantage Actor-Critic algorithm is used to handle multi-agent environments, and Deep Neural Network (DNN) structures are designed to facilitate the algorithm's convergence. Notably, our work is the first to utilize MADRL in determining the optimal joint policy for signal control and vehicle routing. Numerical experiments conducted on the modified Sioux network demonstrate that our integration of signal control and vehicle routing outperforms controlling signal timings or vehicles' routes alone in enhancing traffic efficiency.