A resilient steel frame structure equipped with replaceable ductile and energy-dissipative connection components was developed in this study. Connective components, such as splice plates and T-stubs are made of low-yield-point steels (LYPs), which have superior ductility and energy dissipation capacity. The plastic hinge can be separated from the column surface by local weakening, thus concentrating local damage within low-yield-point steel connection components (LYP-Cs) to effectively enhance ductility of the structure. Five-story and ten-story bolted steel frame structures with four configuration parameters were designed to investigate the seismic performance and damage development of steel frame structures with LYP-Cs (SF-LYP-Cs). Corresponding numerical models were established. Welded steel frame structures with traditional reduced beam sections (SF-RBS) were also established for comparison. Pushover analyses were conducted to explore the effect of structural fuses and to obtain multi-stage performance indices of the resilient steel frame structure based on energy dissipation proportion (EDP) and damage behavior. Incremental dynamic analysis was performed to verify proposed performance indices, and seismic performance of SF-LYP-Cs was evaluated via fragility analysis. The results showed that LYP-Cs were not only effective structural fuses but also effectively delayed the damage development in the primary members, thus reducing the failure probability of the structures. The multi-stage performance indices were conductive to describe the multiple lines of resistance and effects of structural fuses. A larger weakening degree and smaller beam height mitigated plastic damage development in primary members. When LYP-Cs dominate the system’s energy dissipation, the section sizes of columns and the number of structural stories exert only slight influence on the structural performance indices.