Negative autoregulation is universally found across organisms. In the bacterium Escherichia coli, transcription factors often repress their own expression to form a negative feedback network motif that enables robustness to changes in biochemical parameters. Here we present a simple phenomenological model of a negative feedback transcription factor repressing both itself and another target gene. The strength of the negative feedback is characterized by three parameters: the cooperativity in self-repression, the maximal expression rate of the transcription factor, and the apparent dissociation constant of the transcription factor binding to its own promoter. Analysis of the model shows that the target gene levels are robust to mutations in the transcription factor, and that the robustness improves as the degree of cooperativity in self-repression increases. The prediction is tested in the LexA transcriptional network of E. coli by altering cooperativity in self-repression and promoter strength. Indeed, we find robustness is correlated with the former. Considering the proposed importance of gene regulation in speciation, parameters governing a transcription factor's robustness to mutation may have significant influence on a cell or organism's capacity to evolve.