The influence of the uniaxial bias stress on the piezoelectric properties of tetragonal BaTiO3 and PbTiO3 monodomain crystals is modeled in the framework of the phenomenological Landau-Ginzburg-Devonshire theory. It is shown that tensile and compressive stresses, both applied along the spontaneous polarization direction, reduce and enhance the piezoelectric response, respectively. The enhancement effect is due to the flattening of the free-energy profile and the corresponding dielectric softening of crystals, caused by the compressive stress. In BaTiO3 crystals, at temperatures close to the tetragonal-orthorhombic phase transition temperature, the free-energy profile flattening and dielectric softening are the largest along axes perpendicular to the polarization direction, facilitating thus the polarization rotation away from the 001 polar axis. The resulting enhancement of the shear piezoelectric coefficient is directly responsible for the increase of the longitudinal piezoelectric coefficient along the 111 axis. At temperatures deep within the tetragonal phase in BaTiO3, and over the whole ferroelectric region of PbTiO3, the flattening of the free-energy profile and the dielectric softening by compressive stress are the strongest along the polar axis. The resulting enhancement of the longitudinal piezoelectric coefficient is thus the largest along the polar 001 direction. These results, which can be applied to other perovskite crystals, have broad implications.