The effect of glass stability on the yielding transition and mechanical properties of periodically deformed binary glasses is investigated using molecular dynamics simulations. We consider a binary mixture first slowly cooled below the glass transition temperature and then mechanically annealed to deeper energy states via small-amplitude oscillatory shear deformation. We show that upon increasing glass stability, the shear modulus and the yielding peak during startup continuous deformation increase towards plateau levels. It is found that during the strain amplitude sweep, the yielding transition occurs at higher amplitudes and it becomes more abrupt in deeply annealed glasses. The processes of initiation and formation of a shear band are elucidated via the spatiotemporal analysis of nonaffine displacements of atoms. These results are important for thermo-mechanical processing of highly stable amorphous alloys.