In this paper we study the high-dimensional super-resolution imaging problem. Here we are given an image of a number of point sources of light whose locations and intensities are unknown. The image is pixelized and is blurred by a known point-spread function arising from the imaging device. We encode the unknown point sources and their intensities via a nonnegative measure and we propose a convex optimization program to find it. Assuming the device's point-spread function is component-wise decomposable, we show that the optimal solution is the true measure in the noiseless case, and it approximates the true measure well in the noisy case with respect to the generalized Wasserstein distance. Our main assumption is that the components of the point-spread function form a Tchebychev system ($T$-system) in the noiseless case and a $T^*$-system in the noisy case, mild conditions that are satisfied by Gaussian point-spread functions. Our work is a generalization to all dimensions of the work (Eftekhari, Bendory, & Tang, 2021) where the same analysis is carried out in 2 dimensions. We resolve an open problem posed in (Schiebinger, Robeva, & Recht, 2018) in the case when the point-spread function decomposes.