We study the hadronic eta' -> eta pi pi decays within the framework of U(3)(L) circle times U(3)(R) Chiral Perturbation Theory including resonance states and the complete one-loop corrections. The amplitude is projected in partial waves and unitarized by means of the N/D method resumming both the important S- and D-wave pi pi and the subleading S-wave pi eta final-state interactions. The participating scalar multiplet mass and coupling strengths are determined from fits to the Dalitz plot experimental data recently released by the A2 collaboration. As a byproduct of our analysis, the associated Dalitz-plot slope parameters are found to be a = -0.072(7)(stat) (8)(syst), b = -0.052(1)(stat)(2)(syst), d = -0.051(8)(stat)(6)(syst), which lie in the ballpark of the current experimental and theoretical determinations.