We study the behavior of the Wasserstein-$2$ distance between discrete measures $μ$ and $ν$ in $\mathbb{R}^d$ when both measures are smoothed by small amounts of Gaussian noise. This procedure, known as Gaussian-smoothed optimal transport, has recently attracted attention as a statistically attractive alternative to the unregularized Wasserstein distance. We give precise bounds on the approximation properties of this proposal in the small noise regime, and establish the existence of a phase transition: we show that, if the optimal transport plan from $μ$ to $ν$ is unique and a perfect matching, there exists a critical threshold such that the difference between $W_2(μ, ν)$ and the Gaussian-smoothed OT distance $W_2(μ\ast \mathcal{N}_σ, ν\ast \mathcal{N}_σ)$ scales like $\exp(-c /σ^2)$ for $σ$ below the threshold, and scales like $σ$ above it. These results establish that for $σ$ sufficiently small, the smoothed Wasserstein distance approximates the unregularized distance exponentially well.