In order to ensure the structural safety of a liquefied natural gas (LNG) heat exchanger in emergency stop operation process, the strength of a brazed structure with rectangular fins and plates is investigated by means of the finite element method. The microstructure of brazed joints and brazing filler is tested by metallographic examination with a scanning electron microscope (SEM). The results show that the maximum shear stress is the main reason for the structural failure at the brazing seam while the brazed joint is mainly subjected to the maximum normal stress. The peak value of the Von Mises equivalent stress in brazed structure with rectangular fins and plates linearly increases with the HMR pressure when the temperature difference is less than 10 K between HMR and LMR. At the same time, the peak value of Von Mises equivalent stress also increases with the equilibrium temperature and temperature difference between LMR and HMR. The aggregation of the elemental Si in the brazed joints and brazing seam will exacerbate the structural safety of the brazed structure in the emergency stop operation process. The above results provide some constructive guidance for the emergency stop operational process for an LNG heat exchanger.