Observations of astrophysical transients have brought many novel discoveries and provided new insights into the physical processes at work under extreme conditions in the Universe. Multi-wavelength and multi-messenger observations of variable objects require dedicated procedures and follow-up systems capable of digesting and reacting to external alerts to execute coordinated follow-up campaigns. The main functions of such follow-up systems are the processing, filtering, and ranking of the incoming alerts, the fully automated rapid execution of the observations according to an observation strategy tailored to the instrument, and real-time data analysis with feedback to the operators and other instruments. The High Energy Stereoscopic System (H.E.S.S.) has been searching for transient phenomena since its inauguration in 2003. In this paper, we describe the transients follow-up system of H.E.S.S. which became operational in 2016. The system allows H.E.S.S. to conduct a more versatile, optimised, and largely autonomous transient follow-up programme, combining all major functionalities in one systematic approach. We describe the design, central functionalities, and interfaces of the follow-up system in general and its three main components in detail: the Target of Opportunity (ToO) alert system, the data acquisition and central control system, and the real-time analysis. We highlight architectural decisions and features that enable fully automatic ToO follow-up and indicate key performance metrics of the subsystems. We discuss the system's capabilities and highlight the need for a fine-tuned interplay of the different subsystems in order to react quickly and reliably. Lessons learnt from the development, integration, and operation of the follow-up system are reviewed in light of new and large science infrastructures and associated challenges in this exciting new era of inter-operable astronomy.