Using particle swarm optimization and first-principles calculations, the crystal structures, electronic properties, and superconducting behavior of the ternary hydride LuThH₁₀ under high pressure are investigated. Our study uncovers two thermodynamically stable phases with space group symmetries of C2/m and Cmmm. Spectral function and electron-phonon coupling calculations show superconducting transition temperatures (T_c) of 65.8 K and 70.7 K under 200 GPa for the C2/m and Cmmm phases of LuThH₁₀, respectively. At 300 GPa, the T_c of the C2/m phase reaches 60.0 K. Further analysis reveals that the H atoms make critical contributions to the superconducting properties of LuThH₁₀, by which the high-frequency vibration of the H atoms enhances T_c.