Impacts of climate change on individual species are increasingly well documented, but we lack understanding of how these effects propagate through ecological communities. Here we combine species distribution models with ecological network analyses to test potential impacts of climate change on >700 plant and animal species in pollination and seed-dispersal networks from central Europe. We discover that animal species that interact with a low diversity of plant species have narrow climatic niches and are most vulnerable to climate change. In contrast, biotic specialization of plants is not related to climatic niche breadth and vulnerability. A simulation model incorporating different scenarios of species coextinction and capacities for partner switches shows that projected plant extinctions under climate change are more likely to trigger animal coextinctions than vice versa. This result demonstrates that impacts of climate change on biodiversity can be amplified via extinction cascades from plants to animals in ecological networks.