The effect of small additives of functionalized multiwall carbon nanotubes used as a modifier on the formation and properties of epoxy polymers cured with diaminodiphenyl sulfone is investigated. In the range of additive concentrations 0.01-0.50 wt %, there are extreme dependences of dynamic storage modulus and the glass-transition temperature on modifier concentration. As shown by electron microscopy and X-ray scattering, regions with increased packing densities of macromolecules are formed in the polymers in the presence of the modifier. The effect of the specific surface on the kinetics of curing of epoxy resins is observed. A mechanism controlling the formation of the epoxy matrix that is responsible for the inhomogeneous polymer structuring that defines the final properties of the polymers is suggested.