To enable rational design of high entropy alloys (HEAs), we have developed a first principles density functional theory (DFT) computational approach to predict the mechanical properties of HEAs. Specifically, we used the DFT method to calculate the lattice constants, elastic constants, and generalized stacking fault energy (GSFE) for select HEAs with face-centered cubic lattice structure and varying chemical composition. The predicted mechanical properties include Young’s modulus, shear modulus, and yield strength of the HEAs. We have examined our computational approach for four alloy systems, i.e., NiCoFe, CoCrFeNi, CoCrFeCuNi, and RdIrPdPtNiCu HEAs. For these four HEAs with dramatically different chemical composition, our predicted mechanical properties are found to agree well experimental values. Consequently, we have demonstrated that the developed first principles based computational approach is a reliable computational tool for understanding the composition-structure-property relation of HEAs and, particularly, exploring novel HEAs with superior mechanical properties over vast composition space.