This paper proposes a high-performance control technique based on Lyapunov's stability theory for a single-phase grid-connected neutral-point-clamped quasi-impedance source inverter with LCL filter. The Lyapunov function based control is employed to regulate the inverter output current, whereas the proportional resonant controller is used to produce the reference of the inverter output current that is needed in the Lyapunov function based control. Use of proportional resonant controller ensures the zero steady-state error in the grid current. An important feature of the proposed Lyapunov function based control is the achievement of resonance damping without using a dedicated damping method. Furthermore, the modified simple boost control technique is proposed to eliminate the double-line frequency ripples in the quasi-impedance source inductor currents and minimize the double-line frequency ripples in the quasi-impedance source capacitor voltages. The proposed control technique considerably reduces the inverter size, weight, and cost as well as increases overall system efficiency since the required inductances and capacitances sizes are lower. Experimental results obtained from a 2.5 kW neutral-point-clamped quasi-impedance source inverter prototype are presented to validate the performance of the Lyapunov function based control technique.