We demonstrate theoretically and experimentally how magnetic fields influence the single-electron tunneling dynamics in electron pumps, giving a massively enhanced quantization accuracy and providing a route to a quantum current standard based on the elementary charge. The field dependence is explained by two effects: field-induced changes in the sensitivity of tunneling rates to the barrier potential and the suppression of nonadiabatic excitations due to a reduced sensitivity of the Fock-Darwin states to the electrostatic potential. These effects lead to a continued improvement in quantisation accuracy at high field which is important for applications in metrology.