High-harmonic generation (HHG) is widely used for up-conversion of amplified (near) infrared ultrafast laser pulses to short wavelengths. We demonstrate that Ramsey-comb spectroscopy, based on two such pulses derived from a frequency-comb laser, enables us to observe phase effects in this process with a few mrad precision. As a result, we could perform the most accurate spectroscopic measurement based on light from HHG, illustrated with a determination of the 5p6→5p58s[23/2]1 transition at 110 nm in Xe132. We improve its relative accuracy 104 times to a value of 2.3×10-10. This is 3.6 times better than shown before involving HHG, and promising to enable 1S-2S spectroscopy of He+ for fundamental tests.