The elastic properties of an Mg85Zn6Y9 (at.%) alloy single crystal with a long-period stacking-ordered (LPSO) structure, synchronized with periodic enrichment of Zn and Y atoms, were investigated, the properties having remained unclear because of the difficulty in growing large single crystals. Directionally solidified (DS) Mg85Zn6Y9 alloy polycrystals consisting of a single phase of the 18R-type LPSO structure were prepared using the Bridgman technique. For the DS polycrystals, a complete set of elastic constants was measured with resonant ultrasound spectroscopy combined with electromagnetic acoustic resonance, in which the texture formed by the directional solidification was taken into account. By analyzing the elastic stiffness of DS polycrystals on the basis of a newly developed inverse Voigt–Reuss–Hill approximation, the elastic stiffness components of the single-crystalline LPSO phase were determined. It was revealed that the Young’s modulus of the LPSO phase along 〈0001〉 in the hexagonal expression was clearly higher than that along 〈112¯0〉, and the Young’s modulus and shear modulus were clearly higher than those of pure magnesium. These findings were validated by first-principles calculations based on density functional theory. Analyses by first-principles calculations and micromechanics modeling indicated that the long periodicity of the 18R-type stacking structure hardly enhanced the elastic modulus, whereas the Zn/Y-enriched atomic layers, containing stable short-range ordered clusters, exhibited a high elastic modulus, which contributed to the enhancement of the elastic modulus of the LPSO phase in the Mg–Zn–Y alloy.