In structurally disordered solids, some atoms or small groups of atoms are able to quantum mechanically tunnel between two nearly equivalent sites. These atomic tunneling systems have been identified as the cause of various low-temperature anomalies of bulk glasses and as a source of decoherence of superconducting qubits where they are sparsely present in the disordered oxide barrier of Josephson junctions. We demonstrated experimentally that minute deformation of the oxide barrier changes the energies of the atomic tunneling systems, and we measured these changes by microwave spectroscopy of the superconducting qubit through coherent interactions between these two quantum systems. By measuring the dependence of the energy splitting of atomic tunneling states on external strain, we verify a central hypothesis of the two-level tunneling model for disordered solids.