This paper presents a Lyapunov-function based control approach with cascaded proportional-resonant (PR) controller for single-phase grid-tied LCL-filtered quasi-Z-source inverter (qZSI). The proposed control approach guarantees the global stability of the closed-loop system and zero steady-state error in the grid current. The reference values for the inverter current and capacitor voltage are generated by using cascaded connected PR controllers. As a consequence of using PR controllers, the need for performing derivative operations and estimating grid side inductance and capacitance is eliminated which, in turn, achieves zero steady-state error in the grid current. The shoot through control of the qZSI is achieved by the simple boost control technique. Computer simulations are conducted to show that the proposed control approach provides a good steady-state and dynamic performance in achieving the required control objectives.