Ternary NiTiPt is of great interest in the biomedical industry with the potential to replace binary NiTi in implanted devices due to the alloy’s compatibility with medical imaging techniques and higher radiopacity. Cohesive and elastic properties are important for the processability and formability of materials using various hot and cold work methods. First principles simulations using density functional theory are reported here to determine the theoretical cohesive and elastic properties of NiTiPt with particular focus on nickel-rich stoichiometry. It has been found that Ni-rich NiTiPt alloy can form energetically stable cubic phase, but only up to a platinum content of around 18.75 at.%. On the other hand, the titanium-rich composition can keep the symmetry throughout the 0–50 at.% Pt range. Based on the observed trend in the elastic moduli cubic orthorhombic monoclinic phase transition has been suggested for both Ni-rich and Ti-rich compositions. With increasing platinum content the nickel-rich alloy becomes stiffer compared to the binary alloy. In contrast, the titanium-rich alloy exhibits a decrease in the Young’s modulus as the platinum concentration increases. While both compositions suffer from elastic instability beyond certain amount of Pt addition, the ratio between the bulk modulus to the shear modulus indicates that they both should be reasonably ductile within the elastically stable range of Pt concentrations.