The near-field interaction between fluorescent emitters and graphene exhibits rich physics associated with local dipole-induced electromagnetic fields that are strongly enhanced due to the unique properties of graphene. Here, we measure emitter lifetimes as a function of emitter-graphene distance d, and find agreement with a universal scaling law, governed by the fine-structure constant. The observed energy transfer rate is in agreement with a 1/d(4) dependence that is characteristic of two-dimensional lossy media. The emitter decay rate is enhanced 90 times (energy transfer efficiency of ~99%) with respect to the decay in vacuum at distances d ≈ 5 nm. This high energy transfer rate is mainly due to the two-dimensionality and gapless character of the monatomic carbon layer. Graphene is thus shown to be an extraordinary energy sink, holding great potential for photodetection, energy harvesting, and nanophotonics.