We show that the evolution of magnetic fields in a primordial plasma, filled with standard model particles at temperatures T greater than or similar to 10 MeV, is strongly affected by the chiral anomaly-an effect previously neglected. Although reactions, equilibrating left and right electrons, are in thermal equilibrium for T less than or similar to 80 TeV, a left-right asymmetry develops in the presence of strong magnetic fields. This results in magnetic helicity transfer from shorter to longer scales and lepton asymmetry present in the plasma until T similar to 10 MeV, which may strongly affect many processes in the early Universe.