Non-native species can dominate plant communities by competitively displacing native species, or because environmental change creates conditions favourable to non-native species but unfavourable to native species. We need to disentangle these mechanisms so that management can target competitively dominant species and reduce their impacts. Joint-species distribution models (JSDMs) can potentially quantify competitive impacts by simultaneously modelling how species respond to environmental variation and to changes in community composition. We describe a JSDM to model variation in plant cover and show how this can be applied to compositional data to detect dominant competitors that cause other species to decline in abundance. We applied the model to an experiment in an invaded grassy-woodland community in Australia where we manipulated biomass removal (through slashing and fencing to prevent grazing by kangaroos) along a fertility gradient. Non-native species dominated plant cover at high fertility sites in the absence of biomass removal. Results from the JSDM identified three of the 72 non-native plant species (Bromus diandrus, Acetosella vulgaris and especially Avena fatua) as having a strong competitive impact on the community, driving changes in composition and reducing the cover of both native and non-native species, particularly in the absence of grazing. The dominant non-native grasses Bromus diandrus and Avena fatua were among the tallest species in the community and had the greatest impact on shorter-statured species, most likely through competition for light under conditions of high fertility and low grazing. Synthesis. We demonstrate a method to measure competitive impact using a joint-species distribution model, which allowed us to identify the species driving compositional change through competitive displacement, and where on the landscape competitive impacts were greatest. This information is central to managing plant invasions: by targeting dominant non-native species with large competitive impacts, management can reduce impacts where they are greatest. We provide details of the modelling procedure and reproducible code to encourage further application.