We propose a new method to probe the magnetic and electric dipole moments of the τ lepton using precise measurements of the differential rates of radiative leptonic τ decays at high-luminosity B factories. Possible deviations of these moments from the Standard Model values are analyzed in an effective Lagrangian approach, thus providing model-independent results. Analytic expressions for the relevant non-standard contributions to the differential decay rates are presented. Earlier proposals to probe the τ dipole moments are examined. A detailed feasibility study of our method is performed in the conditions of the Belle and Belle II experiments at the KEKB and Super-KEKB colliders, respectively. This study shows that our approach, applied to the planned full set of Belle II data for radiative leptonic τ decays, has the potential to improve the present experimental bound on the τ anomalous magnetic moment. On the contrary, its foreseen sensitivity is not expected to lower the current experimental limit on the τ electric dipole moment.