Grid-connected inverter (GCI) has become the main interface for integrating modern power units, such as distributed energy resources, electric vehicles, microgrids and high voltage direct-current transmission systems. To proceed in this direction, this study presents a novel voltage support control strategy to enhance the reliability and stability of the GCI during unbalanced grid fault conditions. The proposed control strategy simultaneously achieves multiple objectives during grid faults; regulating phase voltages magnitude within pre-defined safety limits, increasing the difference between positive sequence (PS) and negative sequence (NS) voltages, eliminating both active-reactive power oscillations and the DC-link voltage oscillations. Reducing active power oscillations ensure an adjustable control for the DC-link voltage oscillations which result in third-order current harmonic component at the grid side. One of the main contributions to previous studies, reference for reactive power is computed online based on resistive-inductive grid impedance model and reference voltage sequences for the grid support. Another important contribution to existing studies is to supply both the active and reactive powers to the utility grid and load. Detailed mathematical analyses are performed to theoretically describe the behaviour of the proposed control strategy. A comprehensive set of results is presented to confirm the theoretical solutions.