We consider an electron-phonon Hamiltonian in which the electron-phonon interaction occurs through a modification of both the electron on-site energy and the intersite hopping amplitude, i.e., a combination of Holstein and Su-Schrieffer-Heeger (SSH)-like models. It is suggested that this model may apply to certain vibrational degrees of freedom in high-Tc oxides. This Hamiltonian is electron-hole asymmetric and reduces in the infinite-frequency limit to a model for hole superconductivity recently discussed. We solve the Eliashberg equations for the model and examine the effect of electron-hole symmetry breaking on superconductivity when the effective electron-electron interaction is retarded. It is found that superconductivity is enhanced by the electron-hole symmetry-breaking interaction even in the presence of strong retardation. Such an interaction is found to be more effective than an electron-hole symmetric interaction of equivalent magnitude in giving rise to superconductivity over a wide range of parameters for both high and low phonon frequencies. The carrier concentration dependence of the transition temperature in the electron-hole asymmetric model is found to become weaker as the phonon frequency decreases. Possible implications of our results for the understanding of superconductivity in real materials are discussed.