Nanostructures on surfaces can be displaced by applying an electric field or electric currents through a material. This induced mass transport is referred to as electromigration. In this article we show that the anisotropy of diffusion may control the direction of motion of electromigrating nanostructures. For this purpose we study in situ and in real time, by Low Energy Electron Microscopy, the motion of 2D one-atom thick islands or one-atom deep holes on a highly anisotropic surface (reconstructed Si(1 00)). Based on experiments and Kinetic Monte Carlo simulations, we propose a simple analytical model that explains most of the observations. In particular, the direction of motion of the nanostructures depends on the diffusion anisotropy and does not necessarily coincide with the electric field direction. This work opens a way for the manipulation of 2D nanostructures by means of an electric field on anisotropic surfaces.