We propose Quantum Wrapper as a novel networking technology that enables simultaneous control, management, and operation of quantum networks that coexist with classical networks. The quantum wrapper networks enable the transparent and interoperable transportation of quantum wrapper datagrams consisting of quantum payloads and, notably, classical headers, to facilitate the datagram switching without measuring or disturbing the qubits of the quantum payload. Furthermore, quantum wrapper networks can utilize the common network control and management for performance monitoring on the classical header, and infer the quantum channel quality.