In this paper we present a novel exponentially fitted finite elementmethod with triangular elements for the decoupled continuity equations in the drift-diffusion model of semiconductor devices.The continuous problem is first formulated as a variational problem using a weighted inner product. A Bubnov-Galerkinfinite element method with a set of piecewise exponential basis functions is then proposed. The method is shown to be stable and can be regarded asan extension to two dimensions of the well-known Scharfetter-Gummel method.Error estimates for the approximate solution and its associated flux are given. These h-order error bounds depend on some first-order seminorms of the exact solution, the exact fluxand the coefficient function of the convection terms.A method is also proposed for the evaluation of terminal currentsand it is shown that the computed terminal currents are convergent andconservative.