The evolution of ever increasing complex life forms has required innovations at the molecular level in order to overcome existing barriers. For example, evolving processes for cell differentiation, such as epigenetic mechanisms, facilitated the transition to multicellularity. At the same time, studies using gene regulatory network models, and corroborated in single-celled model organisms, have shown that mutational robustness and environmental robustness are correlated. Such correlation may constitute a barrier to the evolution of multicellularity since cell differentiation requires sensitivity to cues in the internal environment during development. To investigate how this barrier might be overcome, we used a gene regulatory network model which includes epigenetic control based on the mechanism of histone modification via Polycomb Group Proteins, which evolved in tandem with the transition to multicellularity. Incorporating the Polycomb mechanism allowed decoupling of mutational and environmental robustness, thus allowing the system to be simultaneously robust to mutations while increasing sensitivity to the environment. In turn, this decoupling facilitated cell differentiation which we tested by evaluating the capacity of the system for producing novel output states in response to altered initial conditions. In the absence of the Polycomb mechanism, the system was frequently incapable of adding new states, whereas with the Polycomb mechanism successful addition of new states was nearly certain. The Polycomb mechanism, which dynamically reshapes the network structure during development as a function of expression dynamics, decouples mutational and environmental robustness, thus providing a necessary step in the evolution of multicellularity.