Cooling microwave resonators to near the quantum ground state, crucial for their operation in the quantum regime, is typically achieved by direct device refrigeration to a few tens of millikelvin. However, in quantum experiments that require high operation power such as microwave-to-optics quantum transduction, it is desirable to operate at higher temperatures with non-negligible environmental thermal excitations, where larger cooling power is available. In this Letter, we present a radiative cooling protocol to prepare a superconducting microwave mode near its quantum ground state in spite of warm environment temperatures for the resonator. In this proof-of-concept experiment, the mode occupancy of a 10-GHz superconducting resonator thermally anchored at 1.02~K is reduced to $0.44\pm0.05$ by radiatively coupling to a 70-mK cold load. This radiative cooling scheme allows high-operation-power microwave experiments to work in the quantum regime, and opens possibilities for routing microwave quantum states to elevated temperatures.