An electrical energy storage device with damping function is a capacitor-cell battery formed of a plurality of capacitor cells connected in either series or parallel. Each of the capacitor cells includes a supercapacitor and a pseudocapacitor. The supercapacitor is a non-polarized capacitor internally having a separator, and the pseudocapacitor is a polarized electrochemical capacitor. The supercapacitor and the pseudocapacitor are electrically connected in parallel. The supercapacitor has a capacity close to or equal to that of the pseudocapacitor. When charging the capacitor-cell battery, the supercapacitor in every capacitor cell produces a polarization effect, so that electrical energy is charged thereinto as a result of voltage. And, due to a potential balance between the supercapacitor and the pseudocapacitor, the electrical energy charged into the supercapacitor is rapidly transformed into electric current that flows into and is stored in the pseudocapacitor.