The electron-capture process plays an important role in the evolution of the core collapse of a massive star that precedes the supernova explosion. In this study, the electron capture on nuclei in stellar environment is described in the relativistic energy density functional framework, including both the finite-temperature and nuclear pairing effects. Relevant nuclear transitions J(pi) = 0(+/-), 1(+/-) , 2(+/- )are calculated using the finite-temperature proton-neutron quasiparticle random-phase approximation with the density-dependent meson-exchange effective interaction DD-ME2. The pairing and temperature effects are investigated in the Gamow-Teller transition strength as well as the electron-capture cross sections and rates for Ti-44 and Fe-56 in the stellar environment. It is found that the pairing correlations establish an additional unblocking mechanism similar to the finite-temperature effects, that can allow otherwise blocked single-particle transitions. Inclusion of pairing correlations at finite temperature can significantly alter the electron-capture cross sections, even up to a factor of 2 for Ti-44, while for the same nucleus electron-capture rates can increase by more than one order of magnitude. We conclude that for the complete description of electron capture on nuclei both pairing and temperature effects must be taken into account.