We report on high-pressure p≤45 GPa resistivity measurements on the perovskite-related mixed-valent compound CsAuBr₃. The compounds high-pressure resistivity can be classified into three regions: For low pressures (p<10 GPa) an insulator to metal transition is observed; between p=10 GPa and 14 GPa the room temperature resistivity goes through a minimum and increases again; and above p=14 GPa a semiconducting state is observed. From this pressure up to the highest pressure of p=45 GPa reached in this experiment, the room-temperature resistivity remains nearly constant. We find an extremely large resistivity reduction between ambient pressure and 10 GPa by more than six orders of magnitude. This decrease is among the largest reported changes in the resistivity for this narrow pressure regime. We show-by an analysis of the electronic band structure evolution of this material-that the large change in resistivity under pressure in not caused by a crossing of the bands at the Fermi level. We find that it instead stems from two bands that are pinned at the Fermi level and that are moving toward one another as a consequence of the mixed-valent to single-valent transition. This mechanism appears to be especially effective for the rapid buildup of the density of states at the Fermi level.