Recent studies have proposed that the bathymetric fabric of the seafloor formed at mid-ocean ridges records rapid (23,000 to 100,000 years) fluctuations in ridge magma supply caused by sealevel changes that modulate melt production in the underlying mantle. Using quantitative models of faulting and magma emplacement, we demonstrate that, in fact, seafloor-shaping processes act as a low-pass filter on variations in magma supply, strongly damping fluctuations shorter than about 100,000 years. We show that the systematic decrease in dominant seafloor wavelengths with increasing spreading rate is best explained by a model of fault growth and abandonment under a steady magma input. This provides a robust framework for deciphering the footprint of mantle melting in the fabric of abyssal hills, the most common topographic feature on Earth.