The sodium (Na) superionic conductor is a key component that could revolutionize the energy density and safety of conventional Na-ion batteries. However, existing Na superionic conductors are primarily based on a single-anion framework, each presenting inherent advantages and disadvantages. Here we introduce a family of amorphous Na-ion conductors (Na₂O₂-MCl_y, M = Hf, Zr and Ta) based on the dual-anion framework of oxychloride. Benefiting from a dual-anion chemistry and with the resulting distinctive structures, Na₂O₂-MCl_y electrolytes exhibit room-temperature ionic conductivities up to 2.0 mS cm^-1, wide electrochemical stability windows and desirable mechanical properties. All-solid-state Na-ion batteries incorporating amorphous Na₂O₂-HfCl₄ electrolyte and a Na_0.85Mn_0.5Ni_0.4Fe_0.1O₂ cathode exhibit a superior rate capability and long-term cycle stability, with 78% capacity retention after 700 cycles under 0.2 C (1C = 120 mA g^-1) at room temperature. The discoveries in this work could trigger a new wave of enthusiasm for exploring new superionic conductors beyond those based on a single-anion framework.