We used first-principles calculations based on the density functional theory (DFT) to investigate the structural, electronic, thermal and elastic properties of the half-Heusler alloy, NiZrSn. For this purpose, two electronic codes were used: full-potential local orbital minimum-basis (FPLO) code and WIEN2k softwarepackage. The calculated lattice constant and energy gap were in agreement with the available experimental and theoretical data. Using the FPLO code, the bulk modulus and its first pressure derivative of the NiZrSn compound were calculated to be 141.014 GPa and 4.2, respectively. Using the WIEN2k code, they were calculated to be 118 GPa and 5.1, respectively. The elastic parameters, including elastic constants (C11, C12 and C44), shear modulus (G), anisotropy factor (A), Young's modulus (Y) and Poisson's ratio, were calculated under higher pressures. The effect of pressure on the density of states, band structure, energy gap and elastic constants were then investigated. Using Cauchy pressure and Pugh's ratio, we demonstrated the brittle nature of NiZrSn under ambient conditions and pressure. There is currently no theoretical or experimental data that displays the effects of pressure on these properties for comparison. We found that the specific heat of NiZrSn is saturated at 98 J/mol K, which is above the Debye temperature.