We present a method of centroiding undersampled point spread functions (PSFs) that may be useful, especially when dithering is not an option. If the profile of the expected PSF is known fairly well through the characterization of the telescope and detector used for observing, one can simulate the undersampled PSF at many positions on a simulated pixel grid. The true centroid positions are known because the PSFs are simulated, and so, one can match up each undersampled PSF image to its true centroid location, thus forming a lookup table. One then assigns the centroid position of an observed PSF to the position associated with the PSF in the lookup table that has the smallest squared residual with respect to the observed PSF. We examine a few PSF sizes and demonstrate that the lookup table provides better centroid positions compared to a traditional curve-fitting algorithm when the PSFs are undersampled, even in the presence of noise. The lookup table can also outperform a traditional curve-fitting algorithm in the case of PSFs with a very low signal-to-noise ratio, even if the PSFs are not undersampled.