With the rise of inverter-based resources (IBRs) within the power system, the control of grid-connected converters (GCCs) has become pertinent due to the fact they interface IBRs to the grid. The conventional method of control for a GCC such as the voltage-sourced converter (VSC) is through a decoupled control loop in the synchronous reference frame. However, this model-based control method is sensitive to parameter changes causing deterioration in controller performance. Data-driven approaches such as machine learning can be utilized to design controllers that are capable of operating GCCs in various system conditions. This work explores a deep learning-based control method for a three-phase grid-connected VSC, specifically utilizing a long short-term memory (LSTM) network for robust control. Simulations of a conventional controlled VSC are conducted using Simulink to collect data for training the LSTM-based controller. The LSTM model is built and trained using the Keras and TensorFlow libraries in Python and tested in Simulink. The performance of the LSTM-based controller is evaluated under different case studies and compared to the conventional method of control. Simulation results demonstrate the effectiveness of this approach by outperforming the conventional controller and maintaining stability under different system parameter changes.