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We prove that the generating functions for the colored HOMFLY-PT polynomials of rational links are specializations of the generating functions of the motivic Donaldson–Thomas invariants of appropriate quivers that we naturally associate with these links. This shows that the conjectural links–quivers correspondence of Kucharski–Reineke–Stošić–Sułkowski as well as the LMOV conjecture holds for all rational links. Along the way, we extend the links–quivers correspondence to tangles and, thus, explore elements of a skein theory for motivic Donaldson–Thomas invariants.