A new scenario for creation of galactic magnetic fields is proposed which is operative at the cosmological epoch of the galaxy formation, and which relies on unconventional properties of dark matter. Namely, it requires existence of feeble but long range interaction between the dark matter particles and electrons. In particular, millicharged dark matter particles or mirror particles with the photon kinetic mixing to the usual photon can be considered. We show that in rotating protogalaxies circular electric currents can be generated by the interactions of free electrons with dark matter particles in the halo, while the impact of such interactions on galactic protons is considerably weaker. The induced currents may be strong enough to create the observed magnetic fields on the galaxy scales with the help of moderate dynamo amplification. In addition, the angular momentum transfer from the rotating gas to dark matter component could change the dark matter profile and formation of cusps at galactic centers would be inhibited. The global motion of the ionized gas could produce sufficiently large magnetic fields also in filaments and galaxy clusters.