The luminescence properties of polar ZnO/(Mg,Zn)O quantum wells (QWs) are determined, besides confinement effects, by a redshift caused by the Stokes shift and the quantum-confined Stark effect (QCSE). We present a comprehensive study of photoluminescence and optical transmission measurements to separate these two effects. Single QW structures have been prepared by pulsed laser deposition on a-plane sapphire exhibiting excitons with and without a distinct QCSE. The QCSE leads to a redshift of the QW luminescence maximum beneath the free exciton energy in ZnO as well as a change of the dynamics from a single exponential decay function to a non-exponential one, well described by a stretched exponential decay function. The internal electric field was evaluated to 0.66 MV/cm. In the absence of an internal electric field, the Stokes shift was determined to drop from 64 meV down to 24 meV with increasing well width. Additionally, with increasing temperature, the QCSE vanishes as the internal electric field is screened by free charge carriers, leading to a drop of the redshift above 150 K to a value comparable to the Stokes shift determined for the QWs without a distinct QCSE. The radiative decay time increases linearly at low temperatures for both types of QWs, indicating free exciton emission as the major recombination channel.