Neutrinos are the most elusive particles in the universe, capable of traveling nearly unimpeded across it. Despite the vast amount of data collected, a long-standing and unsolved issue is still the association of high-energy neutrinos with the astrophysical sources that originate them. Among the candidate sources of neutrinos, there are blazars, a class of extragalactic sources powered by supermassive black holes that feed highly relativistic jets, pointed toward Earth. Previous studies appear controversial, with several efforts claiming a tentative link between high-energy neutrino events and individual blazars, and others putting into question such relation. In this work, we show that blazars are unambiguously associated with high-energy astrophysical neutrinos at an unprecedented level of confidence, i.e., a chance probability of 6 × 10−7. Our statistical analysis provides the observational evidence that blazars are astrophysical neutrino factories and hence, extragalactic cosmic-ray accelerators.