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Designing stabilizing controllers is a fundamental challenge in autonomous systems, particularly for high-dimensional, nonlinear systems that can hardly be accurately modeled with differential equations. The Lyapunov theory offers a solution for stabilizing control systems, still, current methods relying on Lyapunov functions require access to complete dynamics or samples of system executions throughout the entire state space. Consequently, they are impractical for high-dimensional systems. This paper introduces a novel framework, LYapunov-Guided Exploration (LYGE), for learning stabilizing controllers tailored to high-dimensional, unknown systems. LYGE employs Lyapunov theory to iteratively guide the search for samples during exploration while simultaneously learning the local system dynamics, control policy, and Lyapunov functions. We demonstrate its scalability on highly complex systems, including a high-fidelity F-16 jet model featuring a 16D state space and a 4D input space. Experiments indicate that, compared to prior works in reinforcement learning, imitation learning, and neural certificates, LYGE reduces the distance to the goal by 50% while requiring only 5% to 32% of the samples. Furthermore, we demonstrate that our algorithm can be extended to learn controllers guided by other certificate functions for unknown systems.