ABSTRACT Astrophysical surveys rely heavily on the classification of sources as stars, galaxies, or quasars from multiband photometry. Surveys in narrow-band filters allow for greater discriminatory power, but the variety of different types and redshifts of the objects present a challenge to standard template-based methods. In this work, which is part of a larger effort that aims at building a catalogue of quasars from the miniJPAS survey, we present a machine learning-based method that employs convolutional neural networks (CNNs) to classify point-like sources including the information in the measurement errors. We validate our methods using data from the miniJPAS survey, a proof-of-concept project of the Javalambre Physics of the Accelerating Universe Astrophysical Survey (J-PAS) collaboration covering ∼1 deg2 of the northern sky using the 56 narrow-band filters of the J-PAS survey. Due to the scarcity of real data, we trained our algorithms using mocks that were purpose-built to reproduce the distributions of different types of objects that we expect to find in the miniJPAS survey, as well as the properties of the real observations in terms of signal and noise. We compare the performance of the CNNs with other well-established machine learning classification methods based on decision trees, finding that the CNNs improve the classification when the measurement errors are provided as inputs. The predicted distribution of objects in miniJPAS is consistent with the putative luminosity functions of stars, quasars, and unresolved galaxies. Our results are a proof of concept for the idea that the J-PAS survey will be able to detect unprecedented numbers of quasars with high confidence.