In electric power engineering, randomness has generally been understood as exogenous uncertainty that purely hinders operation, planning, and communication goals. This dissertation, in contrast, presents new methods and principles for power engineering that leverage such uncertainty to improve the underlying mathematical and computational tools used to achieve these goals. Two complementary directions organize these chapters. The first argues that the randomness already induced by the challenges of today's power systems can be controlled and understood, asking what worst-case guarantees and what model information can be yielded from such randomness. This includes a suite of concentration inequalities, comprising statistical tools for bounding the fluctuation of electrical quantities of fundamental interest to power engineers. With these tools in hand, we then turn randomness into a controllable probe that reveals which meters to query, how to best predict the behavior of a network model, and how to select control laws behind aggregate electrical load. Inverting this paradigm by intentionally injecting randomness is the second direction. This starts with a vignette showing that dithered, quantized measurements can make a power network topology recoverable from coarse data, yielding a controllable error bound. Then, similar parametric uncertainty is shown to generalize to a much broader range of tasks, culminating in a unification of randomized rounding and differentiable optimization for efficiently solving graph-structured combinatorial optimization problems. A collection of power network topology control problems is then made tractable in the chapters that follow.