Using molecular dynamics simulations, we investigate the effect of uniaxial elastostatic compression on the potential energy, structural relaxation, and mechanical properties of binary glasses. We consider the three-dimensional Kob-Andersen binary mixture, which was initially cooled from the liquid state to the glass phase with a slow rate at zero pressure. The glass was then loaded with a static stress at the annealing temperature during extended time intervals. It is found that the static stress below the yielding point induces large-scale plastic deformation and significant rejuvenation when the annealing temperature is smaller than a fraction of the glass transition temperature. By contrast, aging effects become dominant at sufficiently small values of the static stress and higher annealing temperatures. The mechanical tests after the elastostatic compression have shown that both the elastic modulus and the yield stress decrease in rejuvenated samples, while the opposite trend was observed for relaxed glasses. These results might be useful for the thermomechanical processing of metallic glasses with optimized mechanical properties.