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Abstract Polymer topology, which plays a principal role in the rheology of polymeric fluids, and non‐equilibrium materials, which exhibit time‐varying rheological properties, are topics of intense investigation. Here, composites of circular DNA and dextran are pushed out‐of‐equilibrium via enzymatic digestion of DNA rings to linear fragments. These time‐resolved rheology measurements reveal discrete state‐switching, with composites undergoing abrupt transitions between dissipative and elastic‐like states. The gating time and lifetime of the elastic‐like states, and the magnitude and sharpness of the transitions, are surprisingly decorrelated from digestion rates and non‐monotonically depend on the DNA fraction. These results are modeled using sigmoidal two‐state functions to show that bulk state‐switching can arise from continuous molecular‐level activity due to the necessity for cooperative percolation of entanglements to support macroscopic stresses. This platform, coupling the tunability of topological composites with the power of enzymatic reactions, may be leveraged for diverse material applications from wound‐healing to self‐repairing infrastructure.