We study the shape of the gas-phase mass-metallicity relation (MZR) of a combined sample of presentday dwarf and high-mass star-forming galaxies using IZI, a Bayesian formalism for measuring chemical abundances presented in Blanc et al. (2015). We observe a characteristic stellar mass scale at M * 10 9.5 M , above which the ISM undergoes a sharp increase in its level of chemical enrichment. In the 10 6 − 10 9.5 M range the MZR follows a shallow power-law (Z ∝ M α * ) with slope α = 0.14 ± 0.08. At approaching M * 10 9.5 M the MZR steepens significantly, showing a slope of α = 0.37 ± 0.08 in the 10 9.5 − 10 10.5 M range, and a flattening towards a constant metallicity at higher stellar masses. This behavior is qualitatively different from results in the literature that show a single power-law MZR towards the low mass end. We thoroughly explore systematic uncertainties in our measurement, and show that the shape of the MZR is not induced by sample selection, aperture effects, a changing N/O abundance, the adopted methodology to construct the MZR, secondary dependencies on star formation activity, nor diffuse ionized gas (DIG) contamination, but rather on differences in the method used to measure abundances. High resolution hydrodynamical simulations of galaxies can qualitatively reproduce our result, and suggest a transition in the ability of galaxies to retain their metals for stellar masses above this threshold. The MZR characteristic mass scale also coincides with a transition in the scale height and clumpiness of cold gas disks, and a typical gas fraction below which the efficiency of star formation feedback for driving outflows is expected to decrease sharply.