Sixth-order compact difference schemes for Poisson equations have been widely investigated in the literature. Nevertheless, those methods are all constructed based on knowing the exact values of the derivatives of the source term. Therefore, this drawback mostly prevents their actual applications as the analytic form of the source term is rarely available. In this paper, we propose a sixth-order quasi-compact difference method, without having to know the derivatives of the source term, for solving the 2D and 3D Poisson equations. Our strategy is to discretize the equation by the fourth-order compact scheme at the improper interior grid points that adjoin the boundary, while the sixth-order scheme, where it is compact only for the unknowns, is exploited to the proper interior grid points that are not adjoining the boundary. Theoretically, we rigorously prove that the proposed method can achieve the global sixth-order accuracy. Since there are no derivatives of the source term involved in the proposed scheme, our global sixth-order quasi-compact difference method can be developed to solve the time-dependent problems using a time advancing scheme. Numerical experiments are carried out to demonstrate the convergence order and the efficiency of the proposed methods.