We propose a finite element method to solve nonlinear stochastic diffusion equations with a fluctuating source term to model the effect of thermal fluctuations. One of the main challenges when solving these equations numerically is that the discretization introduces artificial correlations in the stochastic fluctuations of the solution. This makes the nonlinear terms difficult to handle numerically, and the solution difficult to interpret physically. One of the key features of the proposed approach is a linear mapping to transform the finite element solution into an equivalent discrete solution that is free of the artificial correlations introduced by the spatial discretization. The mapping can be applied to any spatial discretization, regardless of the choice of shape function, and including those defined on unstructured meshes. We apply our method to several one-dimensional advection-diffusion and reaction-diffusion problems. The numerical results are in good agreement with the analytical solutions, and capture well the fluctuations characterized by both short and long correlation lengths.