The mechanical properties and internal friction (damping capacity) of Mg–Zn–Y alloys with a long-period stacking ordered (LPSO) structure at different Y/Zn atomic ratios (2/1, 3/2 or 4/3) in cast and extrusion were investigated. It was found that the as-cast Mg–Zn–Y alloys with different Y/Zn atomic ratios possess a single LPSO phase with the same stable 18R-type structure. Among the three alloys, the alloy with 3/2 atomic ratio yields the highest damping capacity in low- and high-strain amplitude stages. Two damping peaks particularly P
1 and P
2 are detected in the Mg–Zn–Y alloy with 3/2 atomic ratio at approximately 108 and 220 °C, respectively. These results may be attributed to few solute atoms in Mg matrix and grain boundaries. In addition, the studied alloy with 3/2 atomic ratio exhibits excellent comprehensive properties in as-cast and as-extruded states; this alloy yields an ultimate tensile strength of 346 MPa and maintains a certain damping capacity (Q−1 > 0.01) in extrusion.