A multi-mesh strategy is employed to reduce the computational costs associated with incorporating correlated uncertainties into structural topology optimization problems. Uncertainty quantification is achieved through an existing perturbation-based method. This method transforms uncertainties in stiffness, such as geometric and material property randomness, into an augmented deterministic topology optimization problem. Although the approach has proven quite efficient in general, the uncertainty sources are tied to the finite element analysis mesh. Computational costs then rise significantly when refining the finite element mesh in the presence of correlated random variables. We therefore propose separating the analysis and uncertainty mesh. The technique is evaluated on problems of expected stiffness with uncertainties in elastic modulus and compared to solutions found using the standard perturbation and Monte Carlo topology optimization algorithms.