We prove the global well-posedness of the stochastic Abelian-Higgs equations in two dimensions. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects. The covariant stochastic objects and their multi-linear interactions are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.