We report atomically detailed molecular dynamics simulations characterizing the interaction of the RAP74 winged helix domain with the intrinsically disordered C-terminal of FCP1. The RAP74-FCP1 complex promotes the essential dephosphorylation of RNA polymerase II prior to initiation of transcription. Although disordered in solution, the C-terminal of FCP1 forms an amphipathic helix when bound to RAP74. Our simulations demonstrate that this interaction also reorganizes and stabilizes RAP74. These simulations illuminate the significance of hydrophobic contacts for stabilizing disordered protein complexes, provide new insight into the mechanism of protein binding by winged helix domains, and also reveal "dynamic fuzziness" in the complex as FCP1 retains significant flexibility after binding. In conjunction with our recent NMR experiments identifying residual structure in unbound FCP1, these simulations suggest that FCP1 loses relatively little conformational entropy upon binding and that the associated coupled folding-binding transition may be less sharp than expected.