The influence of temperature and pressure on the electronic, elastic, structural, and thermodynamic properties of Ni3Al alloy was investigated by performing a first-principles study. The calculated elastic constants, equilibrium lattice constants, and elastic modulus agree well with the recorded theoretical and experimental data. The calculated elastic constants indicate that C11 is more sensitive than C12 and C44 to pressure. The Young’s modulus, bulk modulus, and shear modulus increase with an increase in pressure. The ratio of bulk to shear modulus (B/G) and anisotropy factor A were also analyzed. The Debye temperature was obtained by calculating the elastic constants, and it changed with the change in the pressure. The thermal expansion coefficient, normalized volume, heat capacity, bulk modulus, and Debye temperature Θ were determined and analyzed by using the quasi-harmonic Debye model at pressures of 0–60 GPa and temperatures of 0–1600 K. Finally, the density of states and Mulliken population were investigated and the effect of pressure on these was analyzed.