A single-photon switch is an important element for the building of scalable quantum networks. In the paper, we propose a feasible scheme for efficient single-photon switching. The proposed switch is controlled by a state of a qubit formed by the pair of the lowest levels of a three-level system (qutrit) coupled to a resonator. This resonator-qutrit system comprises a switching unit of the considered setup. For suppression of the Purcell relaxation of the control qubit, the switching unit is embedded into a coupled-resonator array serving as an engineered electromagnetic environment with a bandgap on a qubit transition frequency. We discuss the possible implementation of the considered singlephoton switch on the microwave circuit QED architecture. We demonstrate that high switching contrasts can be attained for the parameters achievable for the state-of-the-art superconducting circuit QED setups.