We study the transport of an interacting Bose-Einstein condensate through a 1D correlated disorder potential. We use for this purpose the truncated Wigner method, which is, as we show, corresponding to the diagonal approximation of a semiclassical van Vleck-Gutzwiller representation of this many-body transport process. We also argue that semiclassical corrections beyond this diagonal approximation are vanishing under disorder average, thus confirming the validity of the truncated Wigner method in this context. Numerical calculations show that, while for weak atom-atom interaction strengths Anderson localization is preserved with a slight modification of the localization length, for larger interaction strengths a crossover to a delocalized regime exists due to inelastic scattering. In this case, the transport is fully incoherent.