Owing to the filter resonance and background harmonics, the current control for the LCL-filtered grid-connected inverter should be carefully designed to ensure stable operation. Prior-art current control methods normally require extra sensors to achieve the damping of the resonance, or sophisticated active damping should be employed. Both increase the overall system cost and complexity. In this context, an improved control strategy is proposed for LCL-filtered inverters. The proposed control method utilizes a novel reduced-order observer in a way that only one current sensor is required for stable operation (i.e., resonance and harmonics are effectively attenuated). More specifically, the reduced-order observer embeds the dynamics of the grid voltage, where the estimated grid information is used for synchronization in a phase-locked loop. The estimated state variables of the observer are then used for the controller design. Furthermore, to achieve active damping and suppress the influence of grid voltage distortions on the current quality, a multi-resonant state-space controller is proposed, where a linear quadratic regulator method is employed to obtain the optimal gain. The simulations and experimental tests are performed on a 3-kW grid-connected inverter system with an LCL filter. The results demonstrate the effectiveness of the proposed method in terms of robust active damping and strong harmonic attenuation, and thus the inverter achieves a good power quality with only one current sensor.