High-fidelity gate operations are essential to the realization of a fault-tolerant quantum computer. In addition, the physical resources required to implement gates must scale efficiently with system size. A longstanding goal of the superconducting qubit community is the tight integration of a superconducting quantum circuit with a proximal classical cryogenic control system. Here we implement coherent control of a superconducting transmon qubit using a single-flux quantum (SFQ) pulse driver cofabricated on the qubit chip. The pulse driver delivers trains of quantized flux pulses to the qubit through a weak capacitive coupling; coherent rotations of the qubit state are realized when the pulse-to-pulse timing is matched to a multiple of the qubit oscillation period. We measure the fidelity of SFQ-based gates to be approximately 95% using interleaved randomized benchmarking. Gate fidelities are limited by quasiparticle generation in the dissipative SFQ driver. We characterize the dissipative and dispersive contributions of the quasiparticle admittance and discuss mitigation strategies to suppress quasiparticle poisoning. These results open the door to integration of large-scale superconducting qubit arrays with SFQ control elements for low-latency feedback and stabilization.