Industry 4.0 has significantly improved the industrial manufacturing scenario in recent years. The Industrial Internet of Things (IIoT) enables the creation of globally interconnected smart factories, where constituent elements seamlessly exchange information. Industry 5.0 has further complemented these achievements, as it focuses on a human-centric approach where humans become part of this network of things, leading to a robust human–machine interaction. In this distributed, dynamic, and highly interconnected environment, functional safety is essential for adequately protecting people and machinery. The increasing availability of wireless networks makes it possible to implement distributed and flexible functional safety systems. However, such networks are known for introducing unwanted delays that can lead to safety performance degradation due to their inherent uncertainty. In this context, the Time-Sensitive Networking (TSN) standards present an attractive prospect for enhancing and ensuring acceptable behaviors. The research presented in this paper deals with the introduction of TSN to implement functional safety protocols for wireless networks. Among the available solutions, we selected Wi-Fi since it is a widespread network, often considered and deployed for industrial applications. The introduction of a reference functional safety protocol is detailed, along with an analysis of how TSN can enhance its behavior by evaluating relevant performance indexes. The evaluation pertains to a standard case study of an industrial warehouse, tested through practical simulations. The results demonstrate that TSN provides notable advantages, but it requires meticulous coordination with the Wi-Fi MAC layer protocol to guarantee improved performance.