Quenching of limit-cycle oscillations (LCOs), either through mutual coupling or external forcing, has attracted wide attention in several fields of science and engineering. However, the simultaneous utilization of these coupling schemes in quenching of LCOs has rarely been studied despite its practical applicability. We study the dynamics of two thermoacoustic oscillators simultaneously subjected to mutual coupling and asymmetric external forcing through experiments and theoretical modeling. We investigate the forced response of both identical and nonidentical thermoacoustic oscillators for two different amplitudes of LCOs. Under mutual coupling alone, identical thermoacoustic oscillators display the occurrence of partial amplitude death and amplitude death, whereas under forcing alone, asynchronous quenching of LCOs is observed at nonresonant conditions. When the oscillators are simultaneously subjected to mutual coupling and asymmetric forcing, we observe a larger parametric region of oscillation quenching than when the two mechanisms are utilized individually. This enhancement in the region of oscillation quenching is due to the complementary effect of amplitude death and asynchronous quenching. However, the forced response of coupled nonidentical oscillators shows that the effect of forcing is insignificant on attaining synchronization and quenching of oscillations in the oscillator that is not directly forced. Finally, we qualitatively capture the experimental results using a reduced-order theoretical model of two Rijke tubes that are coupled through dissipative and time-delay coupling and asymmetrically forced. We believe that these findings offer fresh insights into the combined effects of mutual and forced synchronization in a system of coupled nonlinear oscillators. © 2021 American Physical Society.