Spin transport via magnon diffusion in magnetic insulators is important for a broad range of spin-based phenomena and devices. However, the absence of the magnon equivalent of an electric force is a bottleneck. In this Letter, we demonstrate the controlled generation of magnon drift currents in heterostructures of yttrium iron garnet and platinum. By performing electrical injection and detection of incoherent magnons, we find magnon drift currents that stem from the interfacial Dzyaloshinskii-Moriya interaction. We can further control the magnon drift by the orientation of the magnetic field. The drift current changes the magnon propagation length by up to ±6% relative to diffusion. We generalize the magnonic spin transport theory to include a finite drift velocity resulting from any inversion asymmetric interaction and obtain results consistent with our experiments.