ABSTRACT The evolution of shocks induced by massive stars does not depend only on the ambient magnetic field strength, but also on its orientation. In the present work, the dynamics of a magnetized blast wave is investigated under the influence of both azimuthal and axial ambient magnetic fields. The blast wave is driven by a central source and forms a shell that results from the accumulation of interstellar matter behind the shock front. A similarity form of the ambient magnetic field and a cylindrical geometry of the blast wave are assumed to obtain self-similar solutions. The model is studied separately for both azimuthal and axial magnetic field and applied to stellar wind bubbles and supernova remnants respectively, using 1D numerical simulations. We found that the magnetized blast wave differs from the self-similar case without an ambient magnetic field. The forward shock front goes slower in the azimuthal case and faster in the axial one. For both tangential orientations, the thickness of the shell increases with the magnetic strength. In the azimuthal case, the thermal energy can be converted to magnetic energy near the inner boundary of the shell. Thus, the temperature drops and the magnetic field increases at the tangential discontinuity of the stellar wind bubble. In the axial case of a supernova remnant, the numerical solution always follows a special curve in the parameter space given by the self-similar model.