We investigate the single transverse-spin asymmetry with a $sin(2\phi-\phi_S)$ modulation in the pion-induced Drell-Yan process within the theoretical framework of the transverse momentum dependent (TMD) factorization. The asymmetry is contributed by the convolution of the Boer-Mulders function and the transversity. We adopt the model results for the distributions of the pion meson and the available parametrization for the distributions of the proton to numerically estimate the $sin(2\phi-\phi_S)$ asymmetry in $\pi^-p$ Drell-Yan at the kinematics of COMPASS at CERN. To implement the TMD evolution formalism of the distribution functions, we apply two different parametrizations on the nonperturbative Sudakov form factors associated with the distribution functions of the proton and the pion. It is found that our prediction on the single transverse-spin dependent asymmetry $sin(2\phi-\phi_S)$ as functions of $x_p$, $x_\pi$, $x_F$ and $q_\perp$ is
qualitatively consistent with the recent COMPASS measurement in both sign and magnitude.