The scaling up of quantum computers operating in the microwave domain requires advanced control electronics, and the use of integrated components that operate at the temperature of the quantum devices is potentially beneficial. However, such an approach requires ultralow power dissipation and high signal quality to ensure quantum-coherent operations. Here we report an on-chip device that is based on a Josephson junction coupled to a spiral resonator and is capable of coherent continuous-wave microwave emission. We show that the characteristics of the device accurately follow a theory based on the perturbative treatment of a capacitively shunted Josephson junction as the gain element. The infidelity of typical quantum gate operations due to phase noise of this cryogenic 25 pW microwave source is less than 0.1% up to 10 ms evolution time, which is below the infidelity caused by dephasing in state-of-the-art superconducting qubits. Together with future cryogenic amplitude and phase modulation techniques, our approach may lead to scalable cryogenic control systems for quantum processors.