The phenomena occurring in dielectrics (Al2O3 and SiO2) in the field of high-power soft X-ray radiation are investigated. Experimental studies are carried out at the Angara-5-1 facility, which allows a pulsed radiation power density of the source up to 5 MW/cm2 at the location of the examined samples. It is revealed that charge carriers are generated in the surface layer of the irradiated dielectric, which causes current generation in a circuit including the dielectric. A necessary condition for current generation is the presence of “hot” electrons, which provide surface conductivity. A mathematical model is proposed to describe the process of current generation in dielectrics during exposure to high-power pulsed X-ray radiation. The model is based on the joint solution of the kinetic equations for X-ray photons, photoelectrons, conduction electrons and holes in the valence band and self-consistent Maxwell’s equations. In Maxwell’s equations, instead of the external current of secondary charge carriers, the radiation conduction current is used.