We study the problem of efficiently recovering the matching between an unlabelled collection of $n$ points in $\mathbb{R}^d$ and a small random perturbation of those points. We consider a model where the initial points are i.i.d. standard Gaussian vectors, perturbed by adding i.i.d. Gaussian vectors with variance $σ^2$. In this setting, the maximum likelihood estimator (MLE) can be found in polynomial time as the solution of a linear assignment problem. We establish thresholds on $σ^2$ for the MLE to perfectly recover the planted matching (making no errors) and to strongly recover the planted matching (making $o(n)$ errors) both for $d$ constant and $d = d(n)$ growing arbitrarily. Between these two thresholds, we show that the MLE makes $n^{δ+ o(1)}$ errors for an explicit $δ\in (0, 1)$. These results extend to the geometric setting a recent line of work on recovering matchings planted in random graphs with independently-weighted edges. Our proof techniques rely on careful analysis of the combinatorial structure of partial matchings in large, weakly dependent random graphs using the first and second moment methods.