We report first-principles calculations of local heating in nanoscale junctions formed by a single molecule and a gold point contact. Due to the lower current density and larger heat dissipation, the single molecule heats up less than the gold point contact. We also find that, at zero temperature, a threshold bias Vonset of about 6 mV and 11 mV for the molecule and the point contact, respectively, is required to excite the smallest vibrational mode and generate heat. The latter estimate is in very good agreement with recent experimental results on the same system. At a given external bias V below Vonset, heating becomes noticeable when the background temperature is on the order of ∼ e(Vonset − V )/kB. Above Vonset, local heating increases dramatically with increasing bias but is also considerably suppressed by thermal dissipation into the electrodes. The results provide a microscopic picture of current-induced heat generation in atomic-scale structures.