We present the results of a first search for self-annihilating dark matter in nearby galaxies and galaxy clusters using a sample of high-energy neutrinos acquired in 339.8 days of live time during 2009/10 with the IceCube neutrino observatory in its 59-string configuration. The targets of interest include the Virgo and Coma galaxy clusters, the Andromeda galaxy, and several dwarf galaxies. We obtain upper limits on the cross section as a function of the weakly interacting massive particle mass between 300 GeV and 100 TeV for the annihilation into b (b) over bar, W+(W) over bar (-), tau(+)tau(-), mu(+)mu(-) , and nu(nu) over bar. A limit derived for the Virgo cluster, when assuming a large effect from subhalos, challenges the weakly interacting massive particle interpretation of a recently observed GeV positron excess in cosmic rays.