The surface-stress tensor is an important quantity entering the elastic theory of interaction between atomic steps on solid surfaces. For the clean Si(001) surface the difference between its diagonal elements, the surface stress anisotropy DELTAsigma, comes out by about a factor of two smaller in experiment than compared to the value obtained from previous first-principles calculations. This discrepancy was usually assigned to the presence of surface defects; as other plausible causes one could quote unrealistic atomic surface structures used for the calculations or even a failure of the local-density approximation (LDA) on which the calculations were based. This constituted a serious deficiency in the understanding of the Si(001) surface. We demonstrate that, when first-principles LDA calculations are carried out for the perfect Si(001) surface with the correct character of reconstruction, the calculated and measured DELTAsigma agree within the experimental and numerical uncertainties. We conclude that standard LDA calculations can properly describe elastic properties of the Si(001) surface and that the effect of surface defects on the measured surface stress anisotropy is, at least for the measurements reported in the literature, much smaller than it has been previously anticipated.