An ab initio study of the structural, elastic and vibrational properties of the lutetium gallium garnet (Lu3Ga5O12) under pressure has been performed in the framework of the density functional theory, up to 95 GPa. Pressure dependence of the elastic constants and the mechanical stability are analyzed, showing that the garnet structure is mechanically unstable above 87 GPa. Lattice-dynamics calculations in bulk at different pressures have been performed and compared with Raman scattering measurements of the nanocrystalline Tm(3+)-doped Lu3Ga5O12 up to 60 GPa. The theoretical frequencies and pressure coefficients of the Raman active modes for bulk Lu3Ga5O12 are in good agreement with the experimental data measured for the nano-crystals. The contributions of the different atoms to the vibrational modes have been analyzed based on the calculated total and partial phonon density of states. The vibrational modes have been discussed in relation to the internal and external modes of the GaO4 tetrahedron and the GaO6 octahedron. The calculated infrared modes and their pressure dependence are also reported. Our results show that with this nano-garnet size the sample has essentially bulk properties.