This paper develops weak exponential schemes for the numerical solution of stochastic differential equations (SDEs) with additive noise. In particular, this work provides first and second-order methods which use at each iteration the product of the exponential of the Jacobian of the drift term with a vector. The article also addresses the rate of convergence of the new schemes. Moreover, numerical experiments illustrate that the numerical methods introduced here are a good alternative to the standard integrators for the long time integration of SDEs whose solutions by the common explicit schemes exhibit instabilities.