We report on a spectroscopic study of electronic energy transfer from excitons confined in silicon nanocrystals to triplet ground-state oxygen molecules, being either physisorbed on the nanocrystal surface or present in the gas phase. The broad photoluminescence spectrum of the nanocrystal assembly probes the transfer of excitation and verifies that nonresonant energy transfer proceeds via multiphonon emission. At low temperatures a small spatial separation of the interacting species and a long lifetime of triplet-state excitons provide the strongest coupling. The energy-transfer time to the first and second excited states of molecular oxygen is in the range of 100 mus and shorter than 3 mus, respectively. Nanocrystals with a chemically modified surface are employed to demonstrate that energy transfer is governed by direct electron exchange. Magneto-optical experiments reveal the importance of the spin orientation of the exchanged electrons for the transfer rate. In the regime of intermediate temperatures (110-250 K) the transfer of excitation to the O-2 dimer is resolved.