A large fraction of stars interact with a close companion during their lifetime, in which the transfer of mass and angular momentum shapes their evolution and final fate. Standard rotationally limited accretion models predict that accretors reach critical rotation after gaining only a small fraction of their mass, severely suppressing further accretion. This is in tension with observations of post-interaction systems that require substantial mass gain. We introduce a disc-mediated angular momentum transport prescription for mass-transfer onto stellar companions. This is based on a novel perturbative, analytic star-disc boundary model that allows for continued mass inflow as the accretor approaches critical rotation. Furthermore, the model reproduces previous numerical results in which perturbations to super-critical rotation result in negative torques exerted by the disc, extracting excess angular momentum while allowing continued mass inflow. We implement this mechanism in detailed binary evolution calculations with differential rotation using MESA, and compute grids spanning primary mass, orbital period, and mass ratio. Whereas rotationally limited models predict β_(rm eff)≲ 0.1, the disc model yields sustained mass inflow near critical rotation, with effective mass-transfer efficiencies of β_(rm eff)∼ 0.4-1 across much of the parameter space. The resulting accretor properties are broadly consistent with observed post-interaction sdOB+Be binaries. Disc-mediated angular momentum transport may therefore represent a key missing ingredient in standard binary evolution models, with important implications for rapidly rotating stars and compact-object progenitors.