We experimentally measure and theoretically analyze the three-atom recombination rate, L3, around a narrow s-wave magnetic Feshbach resonance of 6Li−6Li at 543.3 G. By examining both the magnetic field dependence and, especially, the temperature dependence of L3 over a wide range of temperatures from a few μK to above 200 μK, we show that three-atom recombination through a narrow resonance follows a universal behavior determined by the long-range van der Waals potential and can be described by a set of rate equations in which three-body recombination proceeds via successive pairwise interactions. We expect the underlying physical picture to be applicable not only to narrow s wave resonances, but also to resonances in nonzero partial w...