An existing perturbation-based method is extended to consider correlated uncertainties in structural topology optimization problems. The proposed method uses perturbation technique to model uncertainties in the geometry of structures and material properties, and transforms the problem of topology optimization under uncertainty to an augmented deterministic topology optimization problem. This leads to significant computational savings when compared with Monte Carlo-based optimization, which involve multiple formations and inversions of the global stiffness matrix. We study two numerical examples to show the importance of correlation in uncertainty modeling and to verify the proposed method. Numerical examples show that results obtained from the proposed method are in excellent agreement with those obtained when using Monte Carlo-based optimization.