Given $s_1,\dots, s_m \in (n/2, n]$, let $T_{\sigma}$ be a multilinear Fourier multiplier operator associated with a multilinear multiplier $\sigma$ satisfying a Sobolev regularity condition $\sup_{\ell \in \mathbb{Z}}\|\sigma_{\ell }\|_{W^{s_{1},\ldots ,s_{m}}(\mathbb{R}^{mn})}<\infty .$ By the strongly precompactness of Banach space, the authors prove that if $b_{1},\ldots ,b_{m}\in CMO(\mathbb{R}^{n})$, then the commutator $T_{\sigma,\Sigma b}$ is a compact operator from the product Morrey space $L^{p_{1},\lambda }(\mathbb{R}^{n})\times \cdots \times L^{p_{m},\lambda }(\mathbb{R}^{n})$ to the Morrey space $L^{p,\lambda }(\mathbb{R}^{n})$. As an application, the compactness of the commutator $T_{\sigma,\Sigma b}$ from the product Morrey space $L^{p_{1},\lambda }(\mathbb{R}^{n})\times \cdots \times L^{p_{m},\lambda }(\mathbb{R}^{n})$ to the Morrey space $L^{p,\lambda }(\mathbb{R}^{n})$ is also obtained under the Sobolev regularity condition $\sup _{\ell \in \mathbb{Z}}\|\sigma _{\ell }\|_{W^{s}(\mathbb{R}^{mn})}<\infty$ for $s\in (m n/2, m n]$.