Wediscretize the spatially one-dimensional stochastic Landau-Vlasov equation in a simplified form by first-order finite elements on structured quadrilateral grids. We treat the instabilities due to the hyperbolic character of the equation using a least-square stabilization. We use absolutely stable implicit backward Euler scheme for the time integration. We present numerical examples in which we compare exact and numerical solutions to display the accuracy of the proposed scheme.