Chaotic dynamics in optical microcavities, governed dominantly by manifolds, is of great importance for both fundamental studies and photonic applications. Here, we report the experimental observation of a stable manifold characterized by energy and momentum evolution in the nearly chaotic phase space of an asymmetric optical microcavity. By controlling the radius of a fiber coupler and the coupling azimuth of the cavity, corresponding to the momentum and position of the input light, the injected light can in principle excite the system from a desired position in phase space. It is found that once the input light approaches the stable manifold, the angular momentum of the light experiences a rapid increase, and the energy is confined in the cavity for a long time. Consequently, the distribution of the stable manifold is visualized by the output power and the coupling depth to high-$Q$ modes extracted from the transmission spectra, which is consistent with theoretical predictions by the ray model. This work opens a new path to understand the chaotic dynamics and reconstruct the complex structure in phase space, providing a new paradigm of manipulating photons in wave chaos.