We establish a structural correspondence between persistent species sets and organizations of the augmented reaction network under suitable assumptions. Building upon the framework of Chemical Organization Theory, we introduce an augmented reaction network that incorporates advective transport, inflows, and outflows directly into the stoichiometric structure. This formulation allows the persistence analysis of open and spatially distributed systems to be expressed in purely algebraic terms. The main theorem proves that, for every bounded solution of a RDAS, the set of persistent species forms an organization of the augmented network, while the inverse theorem shows that every such organization can be realized as the persistent set of an appropriately constructed RDAS, independently of kinetic details. Together, these results yield a bidirectional correspondence between algebraic organization and dynamical persistence, extending previous diffusion-only results to systems with advection and boundary-driven fluxes. The proposed framework unifies reaction–transport dynamics, boundary exchange, and organizational structure within a single analytical theory. Applications to biological and chemical transport processes, including viral infection dynamics with advective clearance, demonstrate how advection and boundary interactions generate novel distributed organizations and spatially localized persistence patterns. This work establishes the theoretical foundation for structural analysis and inverse design of open reactive media governed by reaction-diffusion-advection equations.