We show how nonclassical correlations in local bipartite states can act as a resource for quantum information processing. Considering the task of quantum random access codes (RACs) through separable Bell-diagonal states, we demonstrate the advantage of superunsteerability over classical protocols assisted with two bits of shared randomness. We propose a measure of superunsteerability which quantifies nonclassicality beyond quantum steering and obtain its analytical expression for Bell-diagonal states in the context of two- and three-setting steering scenarios that are directly related to the quantum and RAC protocols, respectively. The maximal values of our quantifier yield the optimal quantum efficiency for both of the above protocols, thus showing that superunsteerability provides a precise characterization of the nonclassical resource for implementing RACs with separable Bell-diagonal class of states.