We propose a pairing-based method for cooling an atomic Fermi gas. A three-component (labels 1, 2, 3) mixture of fermions is considered where components 1 and 2 interact and, for instance, form pairs whereas component 3 is in the normal state. For cooling, components 2 and 3 are coupled by an electromagnetic field. Since the quasiparticle distributions in the paired and in the normal states are different, the coupling leads to cooling of the normal state even when initially Tpaired ≥ Tnormal. The cooling efficiency is given by the pairing energy and by the linewidth of the coupling field. No superfluidity is required. The method has a conceptual analogy to cooling based on superconductor-normal-metal tunneling junctions. The main differences arise from the exact momentum conservation in the case of the field-matter coupling versus nonconservation of momentum in the solid-state tunneling process. Moreover, the role of processes that relax the energy conservation requirement in the tunneling-e.g., thermal fluctuations of an external reservoir-is now played by the linewidth of the field. The proposed method should be experimentally feasible due to its close connection to the rf spectroscopy of ultracold gases which is already in use.