The emerging paradigm of resource disaggregation enables the deployment of cloud-like services across a pool of physical and virtualized resources, interconnected using a network fabric. This design embodies several benefits in terms of resource efficiency and cost-effectiveness, service elasticity and adaptability, etc. Application domains benefiting from such a trend include cyber-physical systems (CPS), tactile internet, 5G networks and beyond, or mixed reality applications, all generally embodying heterogeneous Quality of Service (QoS) requirements. In this context, a key enabling factor to fully support those mixed-criticality scenarios will be the network and the system-level support for time-sensitive communication. Although a lot of work has been conducted on devising efficient orchestration and CPU scheduling strategies, the networking aspects of performance-critical components remain largely unstudied. Bridging this gap, we propose KuberneTSN, an original solution built on the Kubernetes platform, providing support for time-sensitive traffic to unmodified application binaries. We define an architecture for an accelerated and deterministic overlay network, which includes kernel-bypassing networking features as well as a novel userspace packet scheduler compliant with the Time-Sensitive Networking (TSN) standard. The solution is implemented as tsn-cni, a Kubernetes network plugin that can coexist alongside popular alternatives. To assess the validity of the approach, we conduct an experimental analysis on a real distributed testbed, demonstrating that KuberneTSN enables applications to easily meet deterministic deadlines, provides the same guarantees of bare-metal deployments, and outperforms overlay networks built using the Flannel plugin.