‘Perytons’ are millisecond-duration transients of terrestrial origin, whose frequency-swept emission mimics the dispersion of an astrophysical pulse that has propagated through tenuous cold plasma. In fact, their similarity to FRB 010724 had previously cast a shadow over the interpretation of ‘fast radio bursts’ (FRBs), which otherwise appear to be of extragalactic origin. Until now, the physical origin of the dispersion-mimicking perytons had remained a mystery. We have identified strong out-of-band emission at 2.3–2.5 GHz associated with several peryton events. Subsequent tests revealed that a peryton can be generated at 1.4 GHz when a microwave oven door is opened prematurely and the telescope is at an appropriate relative angle. Radio emission escaping from microwave ovens during the magnetron shut-down phase neatly explains all of the observed properties of the peryton signals. Now that the peryton source has been identified, we furthermore demonstrate that the microwave ovens on site could not have caused FRB 010724. This and other distinct observational differences show that FRBs are excellent candidates for genuine extragalactic transients.