Organo-metal halide perovskites are a promising platform for optoelectronic applications in view of their excellent charge-transport and bandgap tunability. However, their low photoluminescence quantum efficiencies, especially in low-excitation regimes, limit their efficiency for light emission. Consequently, perovskite light-emitting devices are operated under high injection, a regime under which the materials have so far been unstable. Here we show that, by concentrating photoexcited states into a small subpopulation of radiative domains, one can achieve a high quantum yield, even at low excitation intensities. We tailor the composition of quasi-2D perovskites to direct the energy transfer into the lowest-bandgap minority phase and to do so faster than it is lost to nonradiative centers. The new material exhibits 60% photoluminescence quantum yield at excitation intensities as low as 1.8 mW/cm2, yielding a ratio of quantum yield to excitation intensity of 0.3 cm2/mW; this represents a decrease of 2 orders of magnitude in the excitation power required to reach high efficiency compared with the best prior reports. Using this strategy, we report light-emitting diodes with external quantum efficiencies of 7.4% and a high luminescence of 8400 cd/m2.