In this paper, a multiscale problem arising in material science is considered. The problem involves a random coefficient which is assumed to be a perturbation of a deterministic coefficient, in a sense made precisely in the body of the text. The homogenized limit is then computed by using a perturbation approach. This computation requires repeatedly solving a corrector-like equation for various configurations of the material. For this purpose, the reduced basis approach is employed and adapted to the specific context. The authors perform numerical tests that demonstrate the efficiency of the approach.