The authors demonstrate the potential of a methodological approach for monitoring the damage kinetics and evaluating the load-bearing capacity of composite structural elements based on the method of acoustic emission. This methodology was developed by specialists of the Blagonravov Mechanical Engineering Research Institute of the Russian Academy of Sciences. The methodology was implemented for a fatigue test of an aircraft panel represented by a complex hybrid structure with a lining made of layered composite and a cellular internal structure of aluminum foil with polymeric filler that has a continuous layered structure in the gripping area. Fatigue tests were carried out at 4 Hz from the zero-loading cycle with an amplitude of 145 kN. To record the accumulation of damages, R15–AST transducers by Mistral (USA) and an A-line 32D 8-channel acoustic emission system by Interunis-IT LLC were used. Comparing the weight content of location impulses with their threshold values in energy clusters characterizing the kinetics of micro-, meso-, and macro-damages of the composite material structure made it possible to determine the bearing capacity of the aircraft panel at respective stages of damage kinetics. The results of the acoustic emission diagnostics are given for the examined aircraft panel during cyclic loading, including the coordinate location of acoustic emission sources, accumulation of acoustic emission events by channels, the dynamics of partial activity changes, and weight content of location impulses at damage evolution stages of a composite material. Therefore, the acoustic emission diagnostics of the aircraft panel made it possible to identify the areas of intensive damage accumulation in the hybrid structure of the panel and control the actual bearing capacity by defining the extent of its damages at various scale and structural levels during cyclic loading. The described methodology of using acoustic emission diagnostics for monitoring the damage kinetics and actual bearing capacity of highly loaded composite elements extends the research potential of the acoustic emission method.