We provide probabilistic quasar and galaxy classifications for 2.7 million sources in Gaia Data Release 2. This has been achieved using a supervised classification method (Gaussian Mixture Models) based only on photometric and astrometric data (8 features) in Gaia-DR2. The model is trained empirically to classify objects into three classes - star, quasar, galaxy - for all objects with G>=14.5mag down to the Gaia magnitude limit of G=21.0mag. We provide the probabilities for being a quasar (pqso) and a galaxy (pgal); the probability of being a star is pstar = 1-(pqso+pgal), and all other Gaia data can be obtained by cross-matching Gaia-DR2 using the source identifier. As our main goal is to identify extragalactic objects, we only report objects with pqso+pgal>0.5. These probabilities incorporate a sensible class prior, namely that quasars are 500 times rarer than stars, and that galaxies 7500 times rarer than stars. See the paper for details of the purity and completeness of samples drawn from this catalogue, and for more details of its construction, contents, and validation.