Sealing rings are the core components of flange sealing structures and play a crucial role in the storage and operation of gas generators. The aging and deformation of seals affect the safe operation of the device. This paper aims to investigate the effect of rubber aging on the sealing performance of the components, which is realized by nonlinear finite element analysis. Firstly, an accelerated degradation test method was used to obtain the compression permanent deformation and stress–strain curve of rubber during the aging process. A two-dimensional finite element model of the sealing structure was constructed and the Yeoh model was utilized to describe the mechanical response of rubber. During the simulation, the contact area was modified based on the compression permanent deformation, and the Yeoh model was updated based on the stress–strain curve changes obtained by the test. The impact of key parameters such as material property changes, rubber physical section deformation, and fluid pressure on sealing performance during the seal ring aging process was systematically studied. The numerical results indicate that due to the aging deformation of rubber seals, there is a significant decrease in contact stress and contact width, as well as a shift in maximum equivalent stress area. Taking into account these findings, this study proposes a new design concept for sealing structures. This provides a relatively simple research method for studying flange sealing structure performance.