We report on the measurement of the two-neutrino double-β decay of ^{82}Se performed for the first time with cryogenic calorimeters, in the framework of the CUPID-0 experiment. With an exposure of 9.95 kg yr of Zn^{82}Se, we determine the two-neutrino double-β decay half-life of ^{82}Se with an unprecedented precision level, T_{1/2}^{2ν}=[8.60±0.03(stat) _{-0.13}^{+0.19}(syst)]×10^{19} yr. The very high signal-to-background ratio, along with the detailed reconstruction of the background sources allowed us to identify the single state dominance as the underlying mechanism of such a process, demonstrating that the higher state dominance hypothesis is disfavored at the level of 5.5σ.