We study the long-time mixing behavior of the stochastic nonlinear Schrödinger equation in $\mathbb{R}^d$, $d\le 3$. It is well known that, under a sufficiently strong damping force, the system admits unique ergodicity, although the rate of convergence toward equilibrium has remained unknown. In this work, we address the mixing property in the regime of large damping and establish that solutions are attracted toward the unique invariant probability measure at polynomial rates of arbitrary order. Our approach is based on a coupling strategy with pathwise Strichartz estimates.