A kind of graphene-based perfect absorber which can generate low-threshold and high-extinction-ratio optical bistability in the near-IR band is proposed and simulated with numerical methods. The interaction between input light and monolayer graphene in the absorber can be greatly enhanced due to the perfect absorption. The large nonlinear coefficient of graphene and the strong light-graphene interaction contribute to the nonlinear response of the structure, leading to relatively low switching thresholds of less than 2.5 MW/cm2 for an absorber with a Q factor lower than 1000. Meanwhile, the extinction ratio of bistable states in the absorber reaches an ultrahigh value of 47.3 dB at 1545.3 nm. Moreover, the influence of changing the structural parameters on the bistable behaviors is discussed in detail, showing that the structure can tolerate structural parametric deviation to some extent. The proposed bistable structure with ultra-compact size, low thresholds, high extinction ratio, and ultrafast response time could be of great applications for fabricating high-performance all-optical-communication devices.