Energy management strategy (EMS) is a key technology to improve the economy and durability of fuel cell electric vehicles. To overcome the shortcomings of most existing strategies, a new EMS based on quadratic utility function (QUF) is proposed in this paper. Mathematical models of vehicle and components are established, and the accuracy of the models is verified by bench test. QUF is used to decompose the demand power into the output power of fuel cells and batteries, the strategy considers the historical output state and load change rate of the battery and fuel cell. In order to improve the durability of power sources and the economy of the vehicle, the coefficients of the QUF are determined and optimized by using the Nash equilibrium. Simulation results show that the proposed strategy can effectively improve the durability of fuel cells and batteries, reduce hydrogen consumption, and extend driving range. Compared with the FSM strategy, this strategy can reduce the degradation of the batteries by 21.15%. Compared with the fuzzy control strategy, the proposed strategy can reduce the degradation of fuel cells by 36.52%.