This paper resolves an open problem posed in the literature by proving that, in dual-sourcing inventory systems, the optimality gap of tailored base-surge (TBS) policies decays exponentially with the regular source lead time, with the express-source lead time fixed. In contrast to the existing approach, which relies on conditional Jensen inequalities and a vanishing-discount argument to establish asymptotic optimality, we develop a fundamentally different primal-dual framework. On the primal side, we derive a characterization of the optimal TBS policy using convexity and first-order optimality conditions in the TBS parameters (r,S), which we then use to reformulate the TBS objective as a pricing-type representation. On the dual side, we develop infinite-horizon dual formulations for both closed-loop and semi-open-loop problems. A key technical step is verifying a transversality condition to show weak duality for the closed-loop formulations. Finally, we perturb this dual solution to obtain a feasible dual solution for the closed-loop problem; the induced loss in the dual objective gives an explicit upper bound on the TBS optimality gap. By connecting this loss to a GI/GI/1 queue and exploiting its geometric mixing property, we prove the desired exponential decay. This significantly strengthens the state-of-the-art asymptotic-optimality result, which yields only a square root convergence rate.