Although sustainable development is a topic broadly discussed in the literature in relation to existing policy stimulus, a holistic approach to the implementation of sustainability in agriculture—for which there are three dimensions (economic, social and environmental)—is still missing. A regional approach, which averages the entire EU region, could therefore be useful in the long term for recommending directional guidelines for the Common Agricultural Policy (CAP). The objective of this study is to investigate the impact of key groups of CAP instruments and factor intensity on the three above-mentioned aspects of sustainability in the 2004–2017 period, with the assumption that they are all reciprocally related. This goal was achieved by measuring sustainability using the modified sustainable value approach combined with frontier-based nonparametric assessment and applying structural equation modelling, including multilevel random intercept. This research highlights trade-offs between environmental, economic and social efficiency and checks the impact of the EU CAP schemes on the sustainability of environmental, economic, and social dimensions in agriculture. Despite common indications of trade-offs, particularly between economic performance and eco-efficiency, our study shows that in the long term, such feedback has not occurred in any EU regions. Moreover, there are positive interactions between all three dimensions of sustainability from a cross-sectional perspective. The analysis of the impact of CAP subsidies proves that the current system of agri-environmental, set-aside and rural development payments has been effective in the long term, although broader implementation of environmental schemes in regions with lower labour productivity may negatively affect social sustainability.