We show that the method of distributed noise-shaping beta-quantization offers superior performance for the problem of spectral super-resolution with quantization whenever there is redundancy in the number of measurements. More precisely, if the (integer) oversampling ratio $λ$ is such that $\lfloor M/λ\rfloor - 1\geq 4/Δ$, where $M$ denotes the number of Fourier measurements and $Δ$ is the minimum separation distance associated with the atomic measure to be resolved, then for any number $K\geq 2$ of quantization levels available for the real and imaginary parts of the measurements, our quantization method guarantees reconstruction accuracy of order $O(λK^{- λ/2})$, up to constants which are independent of $K$ and $λ$. In contrast, memoryless scalar quantization offers a guarantee of order $O(K^{-1})$ only.