A devastating tsunami struck Palu Bay in the wake of the 28 September 2018 M$_{\mathrm{w}}=7.5$ Palu earthquake (Sulawesi, Indonesia). With a predominantly strike-slip mechanism, the question remains whether this unexpected tsunami was generated by the earthquake itself, or rather by earthquake-induced landslides. In this study we examine the tsunami potential of the co-seismic deformation. To this end, we present a novel geodetic dataset of GPS and multiple SAR-derived displacement fields to estimate a 3D co-seismic surface deformation field. The data reveal a number of fault bends, conforming to our interpretation of the tectonic setting as a transtensional basin. Using a Bayesian framework, we provide robust finite fault solutions of the co-seismic slip distribution, incorporating several scenarios of tectonically feasible fault orientations below the bay. These finite fault scenarios involve large co-seismic uplift (~2 m) below the bay due to thrusting on a restraining fault bend that connects the offshore continuation of two parallel onshore fault segments. With the co-seismic displacement estimates as input we simulate a number of tsunami cases. For most locations for which video-derived tsunami waveforms are available our models provide a qualitative fit to leading wave arrival times and polarity. The modeled tsunamis explain most of the observed runup. We conclude that co-seismic deformation was the main driver behind the tsunami that followed the Palu earthquake. Our unique geodetic dataset constrains vertical motions of the sea floor, and sheds new light on the tsunamigenesis of strike-slip faults in transtensional basins.