We present an ab initio study of the correlation between nuclear matrix elements of neutrinoless double-beta (0νββ) decay and nucleon-nucleon scattering phase shifts in the 1S0 channel. Starting from thirty-four statistically weighted interactions derived from chiral effective field theory, we apply the valence-space in-medium similarity renormalization group to calculate nuclear matrix elements in four key experimental isotopes. Comparing with the 1S0-channel phase shifts given from each interaction, in all cases we observe a strong correlation for scattering energies above 75 MeV. Furthermore, a global sensitivity analysis, enabled by newly developed machine-learning emulators, confirms that the nuclear matrix elements of the decay depend primarily on the C1S0 low-energy constant, which is associated with the phase shifts in that partial wave. These results provide the first clear correlation between 0νββ decay nuclear matrix elements and a measured observable and will therefore serve as a crucial component in ongoing and future refinements of ab initio uncertainty estimates.