We consider Euler-Maxwell equations arising in the modeling of magnetized plasmas. For such equations steady equilibrium states with zero velocity exist. For small initial data, we show global existence of smooth solutions with convergence toward the steady states as the time goes to infinity. In this problem, the main ingredient is an induction argument on the order of the derivatives of solutions in energy estimates. It is also efficient to obtain the global stability of solutions with exponential decay near steady states for Euler-Poisson equations.