The paper presents the results of experimental studies using the acoustic emission method of samples made of fiberglass, the design feature of which is the presence of both base fibers and weft fibers. The test data are presented in relative units and the concept of staged damage accumulation is used, which allows one to recognize subtle phenomena of the nature of the destruction of fiberglass. Tests of the samples with a transverse arrangement of fibers with respect to the applied load showed that the fracture process both in the strain parameters and in the parameters of the total acoustic emission has two stages of damage accumulation: the stage of proportional change of these parameters from stress and the stage of intensive increment of these parameters. In this case, the parameter of the total acoustic emission shows that this process of destruction begins earlier by 5–6% than the strain parameter shows. The proposed methodology and equipment allows us to identify the nature of fracture and assess the tensile strength of such a “staged” composition, and also to solve the problem of what affected the structural unity violations, which may be due to the presence of a set of cracks formed during manufacturing or under the influence of stresses and the external environment during the process of loading