During major magnetic storms, enhanced fluxes of relativistic electrons in the inner magnetosphere have been observed to correlate with ULF waves. The enhancements can take place over a period of several hours. In order to account for such a rapid generation of relativistic electrons, we examine the mechanism of transit-time acceleration of electrons by low-frequency fast-mode MHD waves, here the assumed form of ULF waves. Transit-time damping refers to the resonant interaction of electrons with the compressive magnetic field component of the fast-mode waves via the zero cyclotron harmonic. In terms of quasi-linear theory, a kinetic equation for the electron distribution function is formulated incorporating a momentum diffusion coefficient representing transit-time resonant interaction between electrons and a continuous broadband spectrum of oblique fast-mode waves. Pitch angle scattering is assumed to be sufficiently rapid to maintain an isotropic electron distribution function. It is further assumed that there is a substorm-produced population of electrons with energies of the order of 100 keV. Calculations of the acceleration timescales in the model show that fast-mode waves in the Pc4 to Pc5 frequency range, with typically observed wave amplitudes (ΔB=10--20 nT), can accelerate the seed electrons to energies of order MeV in a period of a few hours. It is therefore concluded that the mechanism examined in this paper, namely, transit-time acceleration of electrons by fast-mode MHD waves, may account for the rapid enhancements in relativistic electron fluxes in the inner magnetosphere that are associated with major storms.