Vector-borne diseases are among those most sensitive to climate because the ecology of vectors and the development rate of pathogens within them are highly dependent on environmental conditions. Bluetongue (BT), a recently emerged arboviral disease of ruminants in Europe, is often cited as an illustration of climate's impact on disease emergence, although no study has yet tested this association. Here, we develop a framework to quantitatively evaluate the effects of climate on BT's emergence in Europe by integrating high-resolution climate observations and model simulations within a mechanistic model of BT transmission risk. We demonstrate that a climate-driven model explains, in both space and time, many aspects of BT's recent emergence and spread, including the 2006 BT outbreak in northwest Europe which occurred in the year of highest projected risk since at least 1960. Furthermore, the model provides mechanistic insight into BT's emergence, suggesting that the drivers of emergence across Europe differ between the South and the North. Driven by simulated future climate from an ensemble of 11 regional climate models, the model projects increase in the future risk of BT emergence across most of Europe with uncertainty in rate but not in trend. The framework described here is adaptable and applicable to other diseases, where the link between climate and disease transmission risk can be quantified, permitting the evaluation of scale and uncertainty in climate change's impact on the future of such diseases.