Existing modeling and control methods for real-world systems typically deal with uncertainty and nonlinearity on a case-by-case basis. We present a universal and robust control framework for the general class of uncertain nonlinear systems. Our data-driven deep stochastic Koopman operator (DeSKO) model and robust learning control framework guarantee robust stability. DeSKO learns the uncertainty of dynamical systems by inferring a distribution of observables. The inferred distribution is used in our robust and stabilizing closed-loop controller for dynamical systems. We also develop a model predictive control framework with integral action to compensate for run-time parametric uncertainty, such as manipulating unknown objects. Modeling and control experiments in simulation show that our presented framework is more robust and scalable for robotic systems than state-of-the-art controllers using deep Koopman operators and reinforcement learning (RL) methods. We demonstrate that our method resists previously unseen uncertainties, such as external disturbances, at a magnitude of up to five times the maximum control input. Furthermore, we test our DeSKO-based control framework on a real-world soft robotic arm. It shows that our framework outperforms model-based controllers that have full knowledge of the model parameters, and the controller can conduct object pick-and-place tasks without further training. Our approach opens up new possibilities in robustly managing internal or external uncertainty while controlling high-dimensional nonlinear systems in a learning framework. This approach serves as a foundation to greatly simplify high-level control and decision-making for robots.