We present a highly accurate single-center three-dimensional numerical integration technique and apply it to atomic structure calculations based on density functional theory. Our integration scheme employs a Stroud-Lebedev formula for spherical integration. For radial integration a division of a radial interval 0 less than or equal to r< infinity into several subregions is adopted and the Gauss-Legendre and the Gauss-Laguerre quadratures are applied to finite and semi-infinite intervals, respectively. The present method can represent the orthonormality of the analytical hydrogen wave functions with 15-figure accuracy at a few hundred integration points per atom. In the atomic structure calculations using numerical basis functions orthonormal integrals and Hamiltonian and dipole matrix elements are calculated with more than IO-figure accuracy. The accuracy of the matrix elements brings the more reliable total energies of atoms.