In self-excited combustion systems, the application of open-loop forcing is known to be an effective strategy for controlling periodic thermoacoustic oscillations, but it is not known whether and under what conditions such a strategy would work on thermoacoustic oscillations that are not simply periodic. In this study, we experimentally examine the effect of periodic acoustic forcing on a prototypical thermoacoustic system consisting of a ducted laminar premixed flame oscillating quasiperiodically on an ergodic
$\mathbb{T}^{2}$
torus at two incommensurate natural frequencies,
$f_{1}$
and
$f_{2}$
. Compared with that of a classical period-1 system, complete synchronization of this
$\mathbb{T}_{1,2}^{2}$
system is found to occur via a more intricate route involving three sequential steps: as the forcing amplitude,
$\unicode[STIX]{x1D716}_{f}$
, increases at a fixed forcing frequency,
$f_{f}$
, the system transitions first (i) to ergodic
$\mathbb{T}_{1,2,f}^{3}$
quasiperiodicity; then (ii) to resonant
$\mathbb{T}_{1,f}^{2}$
quasiperiodicity as the weaker of the two natural modes,
$f_{2}$
, synchronizes first, leading to partial synchronization; and finally (iii) to a
$P1_{f}$
limit cycle as the remaining natural mode,
$f_{1}$
, also synchronizes, leading to complete synchronization. The minimum
$\unicode[STIX]{x1D716}_{f}$
required for partial and complete synchronization decreases as
$f_{f}$
approaches either
$f_{1}$
or
$f_{2}$
, resulting in two primary Arnold tongues. However, when forced at an amplitude above that required for complete synchronization, the system can transition out of
$P1_{f}$
and into
$\mathbb{T}_{1,2,f}^{3}$
or
$\mathbb{T}_{2,f}^{2}$
. The optimal control strategy is to apply off-resonance forcing at a frequency around the weaker natural mode (
$f_{2}$
) and at an amplitude just sufficient to cause
$P1_{f}$
, because this produces the largest reduction in thermoacoustic amplitude via asynchronous quenching. Analysis of the Rayleigh index shows that this reduction is physically caused by a disruption of the positive coupling between the unsteady heat release rate of the flame and the
$f_{1}$
and
$f_{2}$
acoustic modes. If the forcing is applied near the stronger natural mode (
$f_{1}$
), however, resonant amplification can occur. We then phenomenologically model this
$\mathbb{T}_{1,2}^{2}$
thermoacoustic system as two reactively coupled van der Pol oscillators subjected to external sinusoidal forcing, and find that many of its synchronization features – such as the three-step route to
$P1_{f}$
, the double Arnold tongues, asynchronous quenching and resonant amplification – can be qualitatively reproduced. This shows that these features are not limited to our particular system, but are universal features of forced self-excited oscillators. This study extends the applicability of open-loop control from classical period-1 systems with just a single time scale to ergodic
$\mathbb{T}^{2}$
quasiperiodic systems with two incommensurate time scales.