Localised surface plasmons create a non-equilibrium high-energy electron gas in nanostructures that can be injected into other media in energy harvesting applications. Here, we derive the rate of this localised-surface-plasmon mediated generation of hot electrons in nanorods and the rate of injecting them into other media by considering quantum mechanical motion of the electron gas. Specifically, we use the single-electron wave function of a particle in a cylindrical potential well and the electric field enhancement factor of an elongated ellipsoid to derive the energy distribution of electrons after plasmon excitation. We compare the performance of nanorods with equivolume nanoparticles of other shapes such as nanospheres and nanopallets and report that nanorods exhibit significantly better performance over a broad spectrum. We present a comprehensive theoretical analysis of how different parameters contribute to efficiency of hot-electron harvesting in nanorods and reveal that increasing the aspect ratio can increase the hot-electron generation and injection, but the volume shows an inverse dependency when efficiency per unit volume is considered. Further, the electron thermalisation time shows much less influence on the injection rate. Our derivations and results provide the much needed theoretical insight for optimization of hot-electron harvesting process in highly adaptable metallic nanorods.