Allosteric interactions in DNA are crucial for various biological processes. These interactions are quantified by measuring the change in free energy as a function of the distance between the binding sites for two ligands. Here, we show that trends in the interaction energy of ligands binding to DNA can be explained within an elastic birod model, which accounts for the deformation of each strand as well as the change in stacking energy due to perturbations in position and orientation of the bases caused by the binding of ligands. The strain fields produced by the ligands decay with distance from the binding site. The interaction energy of two ligands decays exponentially with the distance between them and oscillates with the periodicity of the double helix in quantitative agreement with experimental measurements. The trend in the computed interaction energy is similar to that in the perturbation of groove width produced by the binding of a single ligand, which is consistent with molecular simulations. Our analysis provides a new framework to understand allosteric interactions in DNA and can be extended to other rod-like macromolecules whose elasticity plays a role in biological functions.