A new method is presented for the modeling of the electron (positron) energy-loss straggling in Monte Carlo transport simulations. First, the Vavilov energy-loss distribution is calculated for electrons and positrons using the Møller and Bhabha collision cross-sections, respectively. The maximum energy transfer in a single collision (E(S)) is considered as variable. Binding effects from low-energy collisions are modeled using the Blunck and Westphal model. Secondly, new algorithms are developed to fit the Vavilov distribution. These algorithms are based on the first three moments of the energy-loss distribution. They are suitable for rapid random sampling of the energy loss. The new algorithms are validated against the Vavilov distribution for electrons and positrons, water and lead, kinetic energy E0 of 0.1, 1, and 10 MeV and several values of E(S) (10, 50, 100, and 200 keV). The developed algorithms are incorporated in a new version of the GEPTS Monte Carlo code called GEPTS(III). Collisions involving energy transfers larger than E(S) are simulated individually and the energy loss due to soft collisions (energy transfers less than E(S)) is sampled using the new algorithms. The straggling effect is therefore taken into account whatever the chosen E(S) value. GEPTS(III) and EGSnrc are used for the calculation of (1) electron dose distributions in water and (2) energy spectra for electrons passing through water and tungsten slabs. Electron beams of 1, 2, 5, 10, and 20 MeV along with varying E(S) values are considered. Electron dose distributions in water are rather insensitive to the soft collision straggling. The use of the new algorithms results in a slight gain in computation time when relatively large E(S) values are used (e.g., E(S) = 1 MeV for 10 MeV electrons). However, the calculation of electron energy spectra is very sensitive to the soft collision straggling. GEPTS(III) (E(S) = 200 keV) is about 5 and 11 times faster than EGSnrc (E(S) = 1 keV) for the case of 2 and 20 MeV electrons passing through 0.025 and 0.25 cm water slabs, respectively. Contrary to EGSnrc, GEPTS(III) accounts for the energy-spectrum broadening due to the binding effects. The resulting differences between the two codes are significant for 5 and 10 MeV electrons passing through a 0.01 cm tungsten slab. Gains in GEPTS(III) computation times (approximately a factor 5) are also observed for tungsten. In short, GEPTS(III) provides significant advantages (rapidity and accuracy) for electron transport simulations, especially those dealing with energy-spectrum calculations, as encountered in clinical electron beam modeling studies. In other respects, the developed approach is more suitable than class-II codes for the use of accurate electron cross sections (numerical data) at low energy (<100 keV).