We present a comprehensive set of simulations that elucidates several features of experimentally observed self-assembled structures in solutions of DNA and mixtures of neutral and cationic lipids. Our simulations are based on the Noguchi-Takasu implicit-solvent coarse-grained model of lipids as rigid trimer molecules [Noguchi, H.; Takasu, M. Phys. Rev. E 2001, 64, 041913.]. This model is extended in our work so that a certain fraction, phi(c), of the lipids carries +1e charge and DNA molecules are introduced as uniformly charged rods. The simplified coarse-grained modeling approach provides a feasible way to study the long time-scale dynamics associated with the evolution of mesoscopically large complexes from initially disordered systems. Our simulations show that, depending on the rigidity parameter kappa(s) which governs the stiffness of the membranes, both lamellar and inverted hexagonal complexes are formed at intermediate charged lipid densities. Disordered structures are formed both when large amounts of neutral lipids (small phi(c)) are introduced which leads to the "erosion" of the spatial order, as well as for large charge densities that result in membrane rupture. A novel phase, where DNA rods and cylindrical micelles form a two-dimensional square lattice analogous to the three-dimensional cubic NaCl-type structure, is observed in the large phi(c) regime for very soft membrane material (kappa(s) = 0).