The invention discloses a fuel cell hybrid power system provided with an elastic energy storage mechanism and a control method of the fuel cell hybrid power system, belongs to the technical field of design and control of fuel cell automobile systems, and solves the problem that the utilization efficiency of a fuel cell of the existing fuel cell hybrid system is pretty low and the braking energy utilization rate is low. The elastic energy storage mechanism is additionally arranged on the basis of the original fuel cell hybrid power system, and braking energy of the power system at the braking or deceleration stage can be effectively reused through switching by adopting the control method; and in addition, braking energy at the deceleration/braking phase of the automobile can be effectively reused by distributing power at the drive phase via a logical gate by adopting the control method, and the fuel cell works in the efficient working area to realize the optimal control of the fuel cell hybrid power vehicle. The fuel cell hybrid power system provided with the elastic energy storage mechanism and the control method of the fuel cell hybrid power system are applicable to energy storage and energy distribution and utilization of the hybrid power system.