This paper presents a new control strategy based on Lyapunov-function and proportional-resonant (PR) controller for single-phase grid-connected LCL-filtered voltage-source inverters (VSIs). While Lyapunov-function-based control guarantees the global stability of the system, the PR controller is employed to process the grid current error and determine the inverter current reference. However, it is shown that the conventional Lyapunov-function-based control (CLFBC) together with the PR control cannot damp the inherent resonance of the LCL filter. Therefore, this control approach is modified by adding a capacitor voltage loop so as to achieve the desired resonance damping. In addition, a transfer function from the reference grid current to actual grid current is formulated in terms of the LCL-filter parameters and their possible variations in the proposed control strategy. An important consequence of using the PR controller is that the need for performing first and second derivative operations in the generation of inverter current reference is eliminated. Also, a zero steady-state error in the grid current is guaranteed in the case of variations in the LCL-filter parameters. The computer simulations and experimental results obtained from a 3.3-kW system show that the proposed control strategy exhibits a good performance in achieving the required control objectives such as fast dynamic response, zero steady-state error, global stability, and sinusoidal grid current with low total harmonic distortion (THD).