The nuclear matrix elements (NMEs) for two-neutrino double-beta decay (2νββ) are studied in the framework of the relativistic nuclear energy density functional. The properties of nuclei involved in the decay are obtained using relativistic Hartree-Bogoliubov model and relevant nuclear transitions are described using the relativistic proton-neutron quasiparticle random phase approximation (pn-ReQRPA). Three effective interactions have been employed, including density-dependent mesonexchange and point coupling interactions, supplemented with nuclear pairing correlations. The 2νββ matrix elements and half-lives are calculated for several nuclides experimentally known to undergo this kind of decay: 48 Ca, 76 Ge, 82 Se, 96 Zr, 100 Mo, 116 Cd, and 128 Te. The model dependence of the NMEs and their sensitivity on the isoscalar pairing strength V0 is investigated, and the optimized value of this parameter is determined. The results of the present study represent an important benchmark for the future applications of the relativistic framework in studies of neutrinoless doublebeta decay.