Decision-making under uncertainty is an important problem that arises in many domains. Achieving robustness guarantees requires well-calibrated uncertainties, which can be difficult to achieve in high-capacity prediction models such as deep neural networks. This paper proposes an end-to-end approach for learning uncertainty-calibrated deep learning models that directly optimizes a downstream decision-making objective with provable robustness. We also propose two concrete applications in energy system operations, including a grid scheduling task as well as an energy storage arbitrage task. As renewable wind and solar generation increasingly proliferate and their variability penetrates the energy grid, learning uncertainty-aware predictive models becomes increasingly crucial for maintaining efficient and reliable grid operation.