The canonical picture of galaxy evolution invokes the injection of energy from supermassive black holes to reduce star formation efficiency in massive galaxies. This process can explain their observed scaling relations, which were already established by cosmic noon (redshifts 2 to 3). However, the physical channels that allow energy and momentum released on sub-pc scales to affect gas on galactic scales are largely unconstrained. Here we report a direct link between quasar dust-reddening and molecular outflows at z ∼ 2.5. By examining the dynamics of warm molecular gas in the inner region of galaxies, we detect outflows with velocities 700–1000 km/s and short timescales of 0.05 Myr that are due to ongoing quasar energy output. We observe outflows only in systems where quasar radiation pressure on dust is sufficiently large to expel their obscuring gas column densities, indicating that radiative feedback regulates gas in the nuclear regions of galaxies. This is in agreement with theoretical models that predict radiation pressure on dust in the vicinity of the black hole is a major driving mechanism of galactic-scale outflows of cold gas. Our findings show that quasar radiation ejects star-forming gas from within nascent stellar bulges at velocities comparable to those seen on larger scales and that molecules survive in outflows even from the most luminous quasars.