A first-order sentence \theta of vocabulary \sigma \cup {S} is successor-invariant in the finite if for every finite \sigma-structure M and successor relations S₁ and S₂ on M, (M,S₁) \models \theta \iff (M, S₂) \models \theta. In this paper I give an example of a non-first-order definable class of finite structures which is, however, defined by a successor-invariant first-order sentence. This strengthens a corresponding result for order-invariance in the finite, due to Y. Gurevich.