Using the first-principles calculations based on density functional theory (DFT), the structural, elastic, electronic, and vibrational properties of LiAl have been explored within the generalized gradient approximation (GGA) using the Vienna ab initio simulation package (VASP). The results demonstrate that LiAl compound is stable in the NaTl-type structure (B32) at ambient pressure, which is in good agreement with the experimental results and there is a structural phase transition from NaTl-type structure (B32) to CsCl-type structure (B2) at around 22.2 GPa pressure value. The pressure effects on the elastic properties have been discussed and the elastic property calculation indicates that the elastic instability could provide a phase transition driving force according to the variations relation of the elastic constant versus pressure. To gain further information about this, we also have investigated the other elastic parameters (i.e., Zener anisotropy factor, Poisson's ratio, Young's modulus, and isotropic shear modulus). The electronic band structure, total and partial density of states, phonon dispersion curves, and one-phonon density of states of B2 and B32 phases are also presented with results.