We applied the recently developed density-functional (DFT) formulation of the electron paramagnetic resonance (EPR) g-tensor to a series of axially symmetric d(1) transition metal complexes (MEX4z-, where M=V, Cr, Mo, W, Tc, and Re; E=O and N; X=F, Cl, and Br). Values for the g-tensor components are determined by an interplay between three contributions arising due to magnetic field-induced coupling between the following orbitals: (a) The singly occupied alpha b(2) ("d(xy)") molecular orbital (alpha-SOMO) and a metal-based vacant d orbital [either b(1) ("d(x)(2)-y(2)") or e(1) ("d(xz)","d(yz)") depending on the tensor component]; (b) the bonding counterparts of the metal's b(1)/e(1)-type d orbitals and the vacant beta-SOMO; and (c) ligand-based occupied MOs (molecular orbitals) of the appropriate symmetry and the beta-SOMO. The first contribution (which is the only term accounted for in the simple ligand field theory) is usually negative, and decreases the g-tensor components relative to the free electron value, while contributions (b) and (c) are positive. Either of the three terms may dominate, so that values both below and above the free electron are obtained naturally. Calculated g tensors exhibit only a moderate dependence on the molecular geometry. Quasi-relativistic VWN (Vosko-Wilk-Nusair) LDA (local density approximation) geometries are in a good agreement with the available experimental data, and are satisfactory for calculation of g tensors. Tensor components obtained with VWN LDA and gradient-corrected BP86 functionals are essentially identical, and always too positive compared to experiment. The residual errors in both components exhibit strong correlation with the position of the transition metal center in the periodic table. Trends in g-tensor components within the same transition row are correctly reproduced by both functionals, so that a simple additive correction brings g(parallel to) and g(perpendicular to) results into a good agreement with experiment. The deficiencies in the calculated g values may be traced back to the overestimation of the covalent character of bonds formed by metal d orbitals in popular approximate functionals. Calculations of EPR g-tensor thus provide a very stringent quality test for approximate density functionals.