We report the application of the new elimination of Rutherford elastic scattering technique for the measurement of proton-induced reaction cross sections utilizing stored ions decelerated to astrophysical energies. This approach results in a background reduction factor of about 1 order of magnitude, enabling the first measurement of a (p, n) cross section in a storage ring. Here, the reaction channels Xe124(p,n) and Xe124(p,γ) have been studied just above the neutron threshold energy. The new data provide valuable constraints for Hauser-Feshbach theory and extrapolation of the (p,γ) cross section to lower energies. Most importantly, for nuclei of limited availability, the method represents a powerful improvement to efficiently study proton-induced reactions at energies within or close to the astrophysical Gamow window, bringing many reaction measurements within reach that were previously inaccessible in the laboratory.