Recently, in the field of periodic nonuniform sampling, researchers (Wang et al., 2019; Asharabi, 2023) have investigated the incorporation of a Gaussian multiplier in the one-dimensional series to improve its convergence rate. Building on these developments, this paper aimed to accelerate the convergence of the two-dimensional periodic nonuniform sampling series by incorporating a bivariate Gaussian multiplier. This approach utilized a complex-analytic technique and is applicable to a wide range of functions. Specifically, it applies to the class of bivariate entire functions of exponential type that satisfy a decay condition, as well as to the class of bivariate analytic functions defined on a bivariate horizontal strip. The original convergence rate of the two-dimensional periodic nonuniform sampling is given by $O(N^{-p})$, where $p\ge 1$. However, through the implementation of this acceleration technique, the convergence rate improved drastically and followed an exponential order, specifically ${\mathrm{e}}^{-\alpha N}$, where $\alpha >0$. To validate the theoretical analysis presented, the paper conducted rigorous numerical experiments.