This study proposes a new type of self-centering damper equipped with novel buckling-restrained superelastic shape memory alloy (SMA) bars. The new solution aims to address some critical issues related to degradation and loss of superelasticity observed in existing tension-only SMA-based self-centering devices, and in addition, to encourage enhanced material utilization efficiency. The cyclic tension-compression behavior of individual SMA bars is experimentally studied first, and subsequently, two proof-of-concept self-centering dampers are manufactured and tested. A simple yet effective numerical model capturing the flag-shaped response of the dampers is then established, and a preliminary system-level analysis is finally conducted to demonstrate the effectiveness of the proposed damper in structural seismic control. The individual SMA bar specimens show asymmetrical flag-shaped hysteretic responses with satisfactory self-centering capability and moderate energy dissipation. Through a specially designed configuration, the proposed damper shows desirable symmetrical and stable hysteretic behavior, and maintains excellent self-centering capability at 6% bar strain. The system-level dynamic analysis indicates that the dampers, as a means of retrofitting, could effectively reduce both the peak and residual inter-story drift ratios of a six-story steel frame. In particular, the mean residual inter-story drift ratio is reduced from over 0.5% to below 0.2% under the maximum considered earthquake, implying elimination of necessary structural realignment even after strong earthquakes.