Residual star formation at late times in early-type galaxies and their progenitors must be suppressed in order to explain the population of red, passively evolving systems we see today. Likewise, residual or newly accreted reservoirs of molecular gas that are fueling star formation must be destroyed. This suppression of star formation in early-type galaxies is now commonly attributed to active galactic nucleus (AGN) feedback wherein the reservoir of gas is heated and expelled during a phase of accretion onto the central supermassive black hole. However, direct observational evidence for a link between the destruction of this molecular gas and an AGN phase has been missing so far. We present new mm-wavelength observations from the IRAM 30 m telescope of a sample of low-redshift SDSS early-type galaxies currently undergoing this process of quenching of late-time star formation. Our observations show that the disappearance of the molecular gas coincides within less than 100 Myr with the onset of accretion onto the black hole and is too rapid to be due to star formation alone. Since our sample galaxies are not associated to powerful quasar activity or radio jets, we conclude that low-luminosity AGN episodes are sufficient to suppress residual star formation in early-type galaxies. This "suppression mode" of AGN feedback is very different from the "truncation mode" linked to powerful quasar activity during early phases of galaxy formation.