The morphology of a rod embedded in a matrix undergoing pinching by interfacial-energy-driven bulk diffusion is determined near the point of pinching. We find a self-similar solution that gives a unique temporal power law and interfacial shape prior to pinching and self-similar solutions after pinching. The theory is compared to experiments that employ in situ four-dimensional X-ray tomographic microscopy for rods of liquid or solid pinching by solute diffusion in the high-diffusivity liquid phase. The excellent agreement between experiment and theory confirms that the interfacial morphology near the singularity is universal both before and after pinching; the shape holds regardless of the material system and initial condition. This also implies that the predictions of the time-dependence of the process can be used to determine the time to pinching for a wide variety of physical systems, and thus provide estimates of the time required for capillarity-driven break-up of microstructures from the detachment of secondary dendrite arms to polymer blends.