We present an effective field theory approach to the topological response of Floquet systems with symmetry group G. This is achieved by introducing a background G gauge field in the Schwinger-Keldysh formalism, which is suitable for far from equilibrium systems. We carry out this program for chiral topological Floquet systems (anomalous Floquet-Anderson insulators) in two spatial dimensions and the group cohomology models of topological Floquet unitaries. These response actions serve as many-body topological invariants for topological Floquet unitaries. The effective action approach also leads us to propose topological response functions that were not considered before. For a particular family of models, we find that our response theory captures a symmetry protected version of the chiral unitary index.