The effects of pressure on the structural, elastic and electronic properties of the intermetallic compound MgRh with CsCl-type structure have been investigated by using first principles calculations from CASTEP program based on the density functional theory (DFT). The computations have been performed using the plane wave pseudopotential method, with local density approximation (LDA) and generalized gradient approximation (GGA). In this work, the properties of MgRh compound have been investigated over a wide range of pressure from 0 to 60 GPa with a step of 15 GPa. The optimized lattice parameters, elastic constants, elastic anisotropy factor, electronic band structure, brittle/ductile behaviour at different pressure for MgRh compound are obtained. We also derived bulk modulus B, and shear modulus G, Young modulus E, poisson's ratio sigma, and the ratio B/G of MgRh compound. Our calculated values are discussed in comparison with previous experimental data and available theoretical values shows good agreement. This is the first theoretical prediction of the elastic, and electronic properties of MgRh compound under different hydrostatic pressure. Finally, the pressure dependence character of density of states and electronic band structure are calculated and analyzed.