The three-level neutral-point-clamped cascaded rectifier (3LNPC-CR) is a key component in power electronic transformers (PET) due to its high efficiency and modular configuration. However, voltage imbalance among submodule DC links may cause system instability and degrade power quality. To address this issue, this paper proposes a voltage balancing strategy based on Model Predictive Control with a Smooth Switching Sequence (MPC-3S). First, a negative-sequence current control strategy is introduced to equalize the voltages among phases. In addition, an improved modulation scheme is developed to predict and optimize system states in real time within the control horizon, dynamically selecting the optimal switching sequence to achieve rapid voltage equalization. Finally, simulation and experimental results on a three-phase, three-module 3LNPC-CR prototype demonstrate that the proposed MPC-3S strategy can achieve fast intra-phase voltage balancing within 0.1 s under load imbalance, while maintaining high-quality grid-side current. These results verify that the proposed method significantly enhances both the dynamic and steady-state performance of 3LNPC-CR systems, providing a practical and efficient solution to the voltage-balancing challenge in PET applications.