Currently, we are faced with an ever-increasing number of devices and objects connected to the Internet aimed at creating the so-called Internet of Things framework, fostering the creation of a connected world of objects. One of the main challenges we are actually facing is constituted by the constrained sizes of such objects: reduced memory, reduced computational capacity, and reduced battery sizes. Particular attention should be devoted to energy efficiency, since a potential energy shortage would negatively impact not only its operation but also network-wide operation, considering the tight connections among any object. According to the 6G system’s use-case related to self-sustainability and zero-energy networks, this paper focuses on an energy-efficient fog network architecture for IoT scenarios, jointly implementing computation offloading operations and simultaneous wireless information and power Transfer (SWIPT), hence, enabling the possibility of jointly transferring energy and computational tasks among the nodes. The system under consideration is composed of three nodes, where an access point (AP) is considered to be always connected to the power network, while a relay node and an end node can harvest energy from the AP. The proposed solution allows to jointly optimize the computation offloading and the energy harvesting phases while maximizing the network lifetime, so as to maximize the operational time of the network. Numerical results obtained on MATLAB demonstrate that the proposed algorithm performs better than the other benchmarks considered for comparison.