Recent advances in novel view synthesis have enabled real-time rendering speeds with high reconstruction accuracy. 3D Gaussian Splatting (3D-GS), a foundational point-based parametric 3D scene representation, models scenes as large sets of 3D Gaussians. However, complex scenes can consist of millions of Gaussians, resulting in high storage and memory requirements that limit the viability of 3D-GS on devices with limited resources. Current techniques for compressing these pretrained models by pruning Gaussians rely on combining heuristics to determine which Gaussians to remove. At high compression ratios, these pruned scenes suffer from heavy degradation of visual fidelity and loss of foreground details. In this paper, we propose a principled sensitivity pruning score that preserves visual fidelity and foreground details at significantly higher compression ratios than existing approaches. It is computed as a second-order approximation of the reconstruction error on the training views with respect to the spatial parameters of each Gaussian. Additionally, we propose a multi-round prune-refine pipeline that can be applied to any pretrained 3D-GS model without changing its training pipeline. After pruning 90 % of Gaussians, a substantially higher percentage than previous methods, our PUP 3D-GS pipeline increases average rendering speed by 3.56× while retaining more salient foreground information and achieving higher image quality metrics than existing techniques on scenes from the Mip-NeRF 360, Tanks & Temples, and Deep Blending datasets. Code and additional visualizations are available at https://pup3dgs.github.io/.