We have carried a detailed theoretical study on structural, elastic, mechanical properties and Debye temperature of cubic (c-Mg2CoH5) and tetragonal (t-Mg2CoH5) structures of magnesium complex hydrides under a maximum external of pressure up to 10 GPa by using first-principles calculations based on density functional theory (DFT). At 0 GPa, the obtained equilibrium lattice parameters for cubic and tetragonal structure of Mg2CoH5 agreed well with the previous experimental and theoretical results. The elastic constants, bulk modulus, shear modulus, B/G ratio, Poisson ratio (ν) and Young’s modulus for the two structures of magnesium complex hydrides have been studied at 0 GPa and under pressure. The results indicate that the external pressure ranging from 0 to 10 GPa can improve the hardness of c- and t-Mg2CoH5 hydrides. The considered hydrides are found mechanically stable with a brittle behaviour with pressure ranging from 0 to 10 GPa. The calculated sound velocities and the Debye temperature goes up with increasing pressure for the two compounds.