The conventional finite control-set model predictive control (FCS-MPC) methods need a cost function with weighting factors to minimize the common-mode voltage (CMV) in the multilevel inverter (MLI) fed electric drive systems. Moreover, these methods require a higher sampling time for real-time implementation, resulting in a rich harmonic content in the inverter ac currents. This article addresses these concerns by proposing a low-complexity FCS-MPC with CMV minimization for a five-level inverter (FLI). The per-phase philosophy is adopted in the design and implementation of the proposed FCS-MPC for an FLI, resulting in a maximum number of predictions of 6 per phase only (a total of 18 predictions in a three-phase FLI system). Moreover, the proposed FCS-MPC minimizes the CMV without using a cost function, leading to superior current harmonic performance. Additionally, Heun's integration method is introduced in the formulation of discrete-time models of the FLI, and they are used in real-time implementation of the proposed FCS-MPC. The superiority of the proposed method is demonstrated through a dSPACE-controlled FLI laboratory prototype. Furthermore, a comparative analysis of the proposed and the conventional FCS-MPC methods is presented in terms of total demand distortion (TDD) of the current, inverter CMV, and the computational burden.