The Earth's largest earthquakes and tsunamis are usually caused by thrust-faulting earthquakes on the shallow part of the subduction interface between two tectonic plates, where stored elastic energy due to convergence between the plates is rapidly released. The tsunami that devastated the Samoan and northern Tongan islands on 29 September 2009 was preceded by a globally recorded magnitude-8 normal-faulting earthquake in the outer-rise region, where the Pacific plate bends before entering the subduction zone. Preliminary interpretation suggested that this earthquake was the source of the tsunami. Here we show that the outer-rise earthquake was accompanied by a nearly simultaneous rupture of the shallow subduction interface, equivalent to a magnitude-8 earthquake, that also contributed significantly to the tsunami. The subduction interface event was probably a slow earthquake with a rise time of several minutes that triggered the outer-rise event several minutes later. However, we cannot rule out the possibility that the normal fault ruptured first and dynamically triggered the subduction interface event. Our evidence comes from displacements of Global Positioning System stations and modelling of tsunami waves recorded by ocean-bottom pressure sensors, with support from seismic data and tsunami field observations. Evidence of the subduction earthquake in global seismic data is largely hidden because of the earthquake's slow rise time or because its ground motion is disguised by that of the normal-faulting event. Earthquake doublets where subduction interface events trigger large outer-rise earthquakes have been recorded previously, but this is the first well-documented example where the two events occur so closely in time and the triggering event might be a slow earthquake. As well as providing information on strain release mechanisms at subduction zones, earthquakes such as this provide a possible mechanism for the occasional large tsunamis generated at the Tonga subduction zone, where slip between the plates is predominantly aseismic.