In this study, first-principles calculations were utilized to investigate the lattice constants, elastic constants, and mechanical properties of gold–copper (Au–Cu) intermetallic compounds (IMCs), including AuCu3, AuCu, and Au3Cu. We also verified the direction dependence of the Young’s modulus, shear modulus, and Poisson’s ratio of the compounds. The calculated lattice parameters agreed with the experimental data, and the single-crystal elastic constants, elastic modulus E, shear modulus G, bulk modulus B, and Poisson’s ratio ν were calculated. For the Young’s and shear moduli, AuCu3 showed the highest anisotropy, followed by AuCu and Au3Cu. The Poisson’s ratios of AuCu3 and Au3Cu crystals were isotropic on (100) and (111) crystal planes and anisotropic on the (110) crystal plane. However, the Poisson’s ratio of the AuCu crystal was anisotropic on (100) and (111) crystal planes and isotropic on the (110) crystal plane.