The spin-orbit entangled (SOE) Jeff-state has been a fertile ground to study novel quantum phenomena. Contrary to the conventional weakly correlated Jeff=1/2 state of 4d and 5d transition metal compounds, the ground state of CuAl2O4 hosts a Jeff=1/2 state with a strong correlation of Coulomb U. Here, we report that surprisingly Cu2+ ions of CuAl2O4 overcome the otherwise usually strong Jahn-Teller distortion and instead stabilize the SOE state, although the cuprate has relatively small spin-orbit coupling. From the x-ray absorption spectroscopy and high-pressure x-ray diffraction studies, we obtained definite evidence of the Jeff=1/2 state with a cubic lattice at ambient pressure. We also found the pressure-induced structural transition to a compressed tetragonal lattice consisting of the spin-only S=1/2 state for pressure higher than Pc=8 GPa. This phase transition from the Mott insulating Jeff=1/2 to the S=1/2 states is a unique phenomenon and has not been reported before. Our study offers a rare example of the SOE Jeff-state under strong electron correlation and its pressure-induced transition to the S=1/2 state.