A problem of control of uncertain 2D overhead crane affected by disturbances with unknown model is considered. An adaptive controller is proposed to solve the above-mentioned problem, which main salient feature is that it ensures asymptotic convergence of errors for both trolley position and payload swing angle not to a bounded set, but to zero without requirement of it a priori knowledge on bounds of the system parameters and disturbance. First of all, the crane is represented in the feedback linearizable form via coordinate transformations under mild assumptions. Then such representation is parameterized in the form of perturbed linear regression equation, which parameters are exactly identified via application of procedure, which has been recently proposed by the authors and is based on the dynamic regressor extension and mixing method and instrumental variables approach. Obtained parameters estimates are substituted into derived ideal control law, and asymptotic stability of the proposed adaptive control system is rigorously proved. All theoretical results are validated via numerical experiments.