The Internet of Things (IoT) has revolutionized connected devices, with applications in healthcare, data analytics, and smart cities. For time-sensitive applications, 5G wireless networks provide ultra-reliable low-latency communication (URLLC) and fog computing offloads IoT processing. Integrating 5G and fog computing can address cloud computing’s deficiencies, but security challenges remain, especially in Authentication and Key Agreement aspects due to the distributed and dynamic nature of fog computing. This study presents an innovative mutual Authentication and Key Agreement protocol that is specifically tailored to meet the security needs of fog computing in the context of the edge–fog–cloud three-tier architecture, enhanced by the incorporation of the 5G network. This study improves security in the edge–fog–cloud context by introducing a stateless authentication mechanism and conducting a comparative analysis of the proposed protocol with well-known alternatives, such as TLS 1.3, 5G-AKA, and various handover protocols. The suggested approach has a total transmission cost of only 1280 bits in the authentication phase, which is approximately 30% lower than other protocols. In addition, the suggested handover protocol only involves two signaling expenses. The computational cost for handover authentication for the edge user is significantly low, measuring 0.243 ms, which is under 10% of the computing costs of other authentication protocols.