Power quality has become a matter of great concern for all energy stakeholders, whether they are network operators, suppliers, producers, or end users. One of the main power quality issues affecting sensitive equipment and loads are short-time voltage variations (disturbances) called voltage sags and swells. This paper focuses on the compensation of voltage disturbances using a novel quasi-Z-Source (qZS) based dynamic voltage restorer (DVR) solution. The proposed DVR is controlled by using a multi-constraint model predictive control (MPC) approach, which regulates the injection of the required compensation voltage in series and synchronism with the grid during the voltage disturbances events. Moreover, as it is mandatory to ensure proper operation of the qZS inverter, the proposed MPC ensures the control of the qZS capacitor voltage at its reference value. Theoretical analysis and simulation results are given to show the high performance of the proposed solution.