Let D be a smooth divisor on a closed K\ahler manifold X. Suppose that Aut_0(D)={Id}. We prove that the Poincar\'e type extremal K\ahler metric with a cusp singularity at D is unique up to a holomorphic transformation on X that preserves D. This generalizes Berman-Berndtsson's work on the uniqueness of extremal K\ahler metrics from closed manifolds to some complete and noncompact manifolds.
