We report a direct lattice QCD calculation of the strange nucleon electric and magnetic form factors. By using high statistics and a variance reduction technique called hierarchical probing, we obtain a clear nonzero signal for both form factors for the first time. We fit the Q^2-dependence and determine the strange magnetic moment as well as the strange electric and magnetic radii. We compare our results to data from parity-violating electron-proton scattering and to other theoretical studies.