In the present manuscript, we investigate the geometric behavior of complete and stochastically complete Cotton solitons isometrically immersed in a generalized Robertson-Walker (GRW) spacetime whose sectional curvature of the Riemannian fiber satisfies suitable curvature constraints. In this configuration, by assuming appropriate hypotheses on the warped structure of the GRW spacetime, we jointly use a Bochner-type formula with several maximum principles, integrability conditions, and a parabolicity criterion in order to show that such a Cotton soliton must be trivial and locally conformally flat. Additionally, applications of our main results for the steady state space, the de Sitter space, the future temporal cone of the Lorentz-Minkowski space, and a specific open subset of the anti-de Sitter space are also presented.