**Abstract** : We develop computational tools for calculating supersymmetric higher-order derivative corrections to eleven-dimensional supergravity using the action principle approach. We show that, provided the superspace Bianchi identities admit a perturbative solution in the derivative expansion, there are at least two independent superinvariants at the eight-derivative order of eleven-dimensional supergravity. Assuming the twelve-superforms associated to certain anomalous Chern-Simons terms are Weil-trivial, there will be a third independent superinvariant at this order. Under certain conditions, at least two superinvariants will survive to all orders in the derivative expansion. However only one of them will be present in the quantum theory: the supersymmetrization of the Chern-Simons terms of eleven dimensional supergravity required for the cancellation of the M5-brane gravitational anomaly by inflow. This superinvariant can be shown to be unique at the eight-derivative order, assuming it is quartic in the fields. On the other hand, a necessary condition for the superinvariant to be quartic is the exactness, in tau-cohomology, of X0,8 -the purely spinorial component of the eight-superform related by descent to the M5-brane anomaly polynomial. In that case it can also be shown that the solution of the Weil-triviality condition of the corresponding twelve-form, which is a prerequisite for the explicit construction of the superinvariant, is guaranteed to exist. We prove that certain highly non-trivial necessary conditions for the tau-exactness of X0,8 are satisfied. Moreover any potential superinvariant associated to anomalous Chern-Simons terms at the eight-derivative order must necessarily contain terms cubic or lower in the fields.