Quasiperiodic oscillations can occur in nonequilibrium systems where two or more frequency components are generated simultaneously. Many studies have explored the synchronization of periodic and chaotic oscillations; however, the synchronization of quasiperiodic oscillations has not received much attention. This study experimentally documents forced synchronization of the quasiperiodic state and the internally locked state of a thermoacoustic oscillator system. This system consists of a gas-filled resonance tube with a nonuniform cross-sectional area. The thermoacoustic oscillator was designed and built in such a way that nonlinear interactions between the fundamental acoustic oscillation mode and the third mode of the gas column are controlled by a temperature difference that is locally created in the resonance tube. Bifurcation diagrams were mapped out by changing the forcing strength and frequency. Separated Arnold tongues were found and both modes were entrained to the external force through complete synchronization. A saddle-node bifurcation was observed in the route from partial to complete synchronization when the forcing strength was relatively weak. However, a Hopf (torus-death) bifurcation was observed when the forcing was relatively strong. In the internally locked state, the bifurcation occurred after the internal locking was broken down by the external force.