The authors report the first application of the Schwinger variational principle to electron-molecule scattering. Results for electron-H2 scattering in the static-exchange approximation show that the Schwinger method can provide accurate solutions of the scattering problem with small discrete basis sets. The Schwinger variational expression is found to converge far more quickly with respect to the size of the basis than any other algebraic expansion technique considered to date. Results are also presented for hybrid trial scattering wave functions containing both continuum and discrete basis functions.