Mott transitions induced by strong electric fields are receiving growing interest. Recent theoretical proposals have focused on the Zener dielectric breakdown in Mott insulators. However, experimental studies are still too scarce to conclude about the mechanism. Here we report a study of the dielectric breakdown in the narrow-gap Mott insulators GaTa4Se(8-x)Te(x). We find that the I-V characteristics and the magnitude of the threshold electric field (Eth) do not correspond to a Zener breakdown, but rather to an avalanche breakdown. Eth increases as a power law of the Mott-Hubbard gap (Eg), in surprising agreement with the universal law Eth is proportional to Eg(2.5) reported for avalanche breakdown in semiconductors. However, the delay time for the avalanche that we observe in Mott insulators is over three orders of magnitude greater than in conventional semiconductors. Our results suggest that the electric field induces local insulator-to-metal Mott transitions that create conductive domains that grow to form filamentary paths across the sample.