Abstract: As an extension of the cloud computing paradigm, fog computing has attracted wide attention due to its advantages of low energy consumption, short time delay, and high bandwidth saving. Meanwhile, the fog computing-based computation offloading mechanism provides strong support for alleviating the pressure of data processing, realizing low delay service, and prolonging the network lifetime. To construct a green and long lifetime Internet of Things (IoT), this paper proposes a fairness and energy co-aware computation offloading scheme for fog-assisted IoT. Based on the joint optimization consideration of the fog node’s computing capacity, bandwidth resource, and offloading decision with energy consumption fairness, an optimization problem is first formulated to minimize the total energy consumption of all computation tasks. Second, a momentum gradient and coordinate collaboration descent-based fair energy minimization algorithm are proposed to solve the above mixed integer nonlinear programming problem. In this algorithm, based on the historical average energy consumption, distance, computing capacity, and residual energy of the fog node, a fair index is designed to obtain the offloading decision with the optimal energy consumption fairness. Minimization of the total energy consumption for processing all tasks can be achieved by jointly optimizing the occupation ratios of computing and bandwidth resources with the developed momentum gradient and coordinate collaboration descent method. Finally, simulation results show that the proposed scheme can achieve a faster convergence speed. Meanwhile, the total energy consumption of this scheme is the lowest compared to the random selection and greedy task offloading (GTO) schemes, the energy consumption fairness of the fog node is the highest, and the network lifetime is enhanced by 23.6% and 31.2% on average, respectively. Furthermore, this scheme can still maintain its performance advantage under different numbers of fog nodes and different task sizes, indicating the high robustness of the proposed scheme.