First-principles and model calculations show that the Dirac surface state of the topological insulator Bi₂Te₃ survives upon moderate Mn doping of the surface layers but can lose its topological character as a function of magnetization direction. The dispersion depends considerably on the direction of the Mn magnetization: for perpendicular magnetization, a gap of 16 meV opens up at the Dirac point; for in-plane magnetization, a tiny gap can be opened or closed in dependence on the magnetization azimuth. The ground state is ferromagnetic, with a critical temperature of 12 K. The results provide a path towards a magnetic control of the topological character of the Dirac surface state and its consequences to spindependent transport properties.