One of the key targets of the upcoming 5G system is to build a mobile network architecture that supports not only classical mobile broadband applications (i.e., Internet and IMS), but also vertical industry services, such as those of automotive systems, e-health, public safety, and smart grid. Vertical industry is known to have specific needs that cannot be sustained by the current cellular networks. More notably, automotive systems require strict quality of service in terms of ultrashort latency for vehicle-to-infrastructure/network (V2I/N) communications. In this paper, we introduce the Follow Me edge-Cloud (FMeC) concept, leveraging the mobile edge computing (MEC) architecture to sustain requirements of the 5G automotive systems. Assuming that automotive services are deployed on MEC entities, FMeC ensures low-latency access to these services by guaranteeing that vehicles (i.e., as well as user equipment on board vehicles) always connect to nearest automotive service. Besides the FMeC architecture, our contribution in this paper consists in presenting a projection of the FMeC solution on an automated driving use case that integrates automotive and Telco infrastructures, to realize the vision of future 5G automotive systems. We introduce the envisioned software defined networking/OpenFlow-based architecture and our mobility-aware framework based on a set of building blocks that permit achieving the automated driving requirements within 5G network. The evaluation results, obtained conjointly through theoretical analysis and computer simulation, show that our proposed solution outperforms baseline approaches in meeting the automated driving latency requirement and minimizing the incurred global cost.