Using superelastic shape memory alloy (SMA) bars and non-asbestos organic friction material, this study developed an innovative self-centering friction damper (SCFD). This hybrid passive control device consists of the self-centering device using SMA bars and the friction energy dissipation device which can provide excellent self-centering ability and energy-absorbing ability to meet the requirements of civil engineering applications. To explore the feasibility and hysteretic properties of the SCFD, experimental tests under cyclic loading were conducted. According to the experimental results, the proposed SCFD exhibited a stable and repeatable flag-shaped hysteretic response, which can achieve the recovered displacement of 76.07% and dissipated energy of 6.04 kJ at 42 mm. The finite element model of the SCFD using ABAQUS software was established and validated by experimental results. And a series of numerical simulations with different parameters were performed, which enables a more in-depth interpretation of the SCFD. Additionally, a system-level nonlinear time-history analysis was performed on a three-story steel frame equipped with and without SCFDs. The dynamic analysis results indicated that the SCFDs could effectively reduce structural damage and enhance post-earthquake recoverability under rare earthquakes.