We present a model of DNA electrophoresis in unentangled polymer solutions based on a new separation mechanism in which the DNA drags along polymer molecules it encounters during migration. Taking into account the deformation and the hydrodynamic resistance of the polymers in the flow, the mutual disengagement time of the DNA and the polymer, and the average number of polymers dragged by one DNA, we build a self-consistent theory leading to predictions for the DNA velocity as a function of the experimental conditions. Our results agree with the data of Barren et al. (1994), and important separation regimes are also identified.