Coarse-graining of atomistic force fields allows us to investigate complex biological problems, occurring at long timescales and large length scales. In this work, we have developed an accurate coarse-grained model for double-stranded DNA chain, derived systematically from atomistic simulations. Our approach is based on matching correlators obtained from atomistic and coarse-grained simulations, for observables that explicitly enter the coarse-grained Hamiltonian. We show that this requirement leads to equivalency of the corresponding partition functions, resulting in a one-step renormalization. Compared to prior works exploiting similar ideas, the main novelty of this work is the introduction of a highly compact set of Hamiltonian basis functions, based on molecular interaction potentials. We demonstrate that such compactification allows us to reproduce many-body effects, generated by one-step renormalization, at low computational cost. In addition, compact Hamiltonians greatly increase the likelihood of finding unique solutions for the coarse-grained force-field parameter values. By successfully applying our molecular renormalization group coarse-graining technique to double-stranded DNA, we solved, for the first time, a long-standing problem in coarse-graining polymer systems, namely, how to accurately capture the correlations among various polymeric degrees of freedom. Excellent agreement is found among atomistic and coarse-grained distribution functions for various structural observables, including those not included in the Hamiltonian. We also suggest higher-order generalization of this method, which may allow capturing more subtle correlations in biopolymer dynamics.