Shape memory alloy corrugated gaskets (SMA-CGs) can adapt to fluctuating working conditions due to their pseudoelasticity (PE) and shape memory effect (SME), which make them excellent sealing components. In this study, the deformation mechanism of SMA-CGs was examined according to their structural properties under installation and operating conditions to establish an SMA-CG thermal-mechanical coupling model with the finite element analysis (FEA) method, which has been verified through experimentation. Based on this, a thermal-mechanical coupling FEA model was built for a bolted flange joint with SMA-CG. The influence of the SMA-CG structure parameters on compression-rebound mechanical properties was also studied under installation and operating conditions. The conclusions are as follows: a thermal-mechanical coupling finite element analysis method was established for NiTi alloy corrugated structures. Through comparison with the experimental results, the maximum error of the maximum compression load was 5.78%, the maximum error of the rebound rate was 8.85%, and the maximum error of the maximum compaction force in the heat recovery stage was 12.2%, all of which were within the <15% acceptable error range of engineering fields. According to the related experiments and finite element results, the maximum compressive force of gasket thermal recovery after unloading was not less than 40% of the initial maximum compressive load. The application of shape memory alloy to corrugated gasket significantly improved its ability in coping with fluctuating load temperatures. The contact pressure of corrugation increased with the increase of sheet thickness (T) and corrugated gasket height (H) under installation and operating conditions, showing a decreasing trend with the increase of pitch (P), of which the order of factors influencing the average contact pressure of corrugated gasket was sheet T > H > P, and when structural parameters of SMA-CG were T = 0.6 mm, H = 4 mm, and P = 2.5 mm, the contact pressure of corrugated gasket was the highest under operating conditions.