The use of an external transverse magnetic field to trigger and to control electron self-injection in laser- and particle-beam driven wakefield accelerators is examined analytically and through full-scale particle-in-cell simulations. A magnetic field can relax the injection threshold and can be used to control main output beam features such as charge, energy, and transverse dynamics in the ion channel associated with the plasma blowout. It is shown that this mechanism could be studied using state-of-the-art magnetic fields in next generation plasma accelerator experiments.