Space-geodetic measurements of surface deformation produced by the most recent eruptions at Fernandina – the most frequently erupting volcano in the Galápagos Archipelago – reveal that all have initiated with the intrusion of subhorizontal sills from a shallow magma reservoir. This includes eruptions from fissures that are oriented both radially and circumferentially with respect to the summit caldera. A Synthetic Aperture Radar (SAR) image acquired 1–2 h before the start of a radial fissure eruption in 2009 captures one of these sills in the midst of its propagation toward the surface. Galápagos eruptive fissures of all orientations have previously been presumed to be fed by vertical dikes, and this assumption has guided models of the origin of the eruptive fissure geometry and overall development of the volcanoes. Our findings allow us to reinterpret the internal structure and evolution of Galápagos volcanoes and of similar basaltic shields. Furthermore, we note that stress changes generated by the emplacement of subhorizontal sills feeding one type of eruption may control the geometry of subsequent eruptive fissures. Specifically, circumferential fissures tend to open within areas uplifted by sill intrusions that initiated previous radial fissure eruptions. This mechanism provides a possible explanation for the pattern of eruptive fissures that characterizes all the western Galápagos volcanoes, as well as the alternation between radial and circumferential fissure eruptions at Fernandina. The same model suggests that the next eruption of Fernandina will be from a circumferential fissure in the area uplifted by the 2009 sill intrusion, just southwest of the caldera rim.