A generalized Scharfetter-Gummel method is proposed to construct the numerical flux for one-dimensional drift-diffusion equations. Instead of taking a constant approximation of the flux as Scharfetter and Gummel did in [1], we consider a p-degree polynomial with p ≥ 1. The high order moments of the approximating flux function serve as intermediaries to bring numerical correction to the Scharfetter-Gummel flux, that the other end turns out to be the solution derivatives. Therefore, local solution reconstructions are required. The resulting schemes are high order and discretize at the same time the convective and diffusive fluxes without having to employ separately different methods to do so. The new schemes with p = 1 and p = 2 are employed to simulate atmospheric pressure discharge where they are applied to the continuity equations for electrons and ions, and solved simultaneously with Poisson's equation. Numerical results indicate that our method are robust and highly accurate.