Suppressing laser phase noise is crucial for advancing phase-sensitive precision measurements, including gravitational wave detection, optical atomic clocks, laser radar, and high-precision optical communication. However, conventional feedback control systems combined with highfinesse optical cavities are fundamentally constrained by quantum shot noise and thermal Brownian noise of the cavity mirrors, which impose an ultimate classical noise-reduction bottleneck even after extensive optimization of cavity length, materials, and cryogenic operation. To overcome this limitation by exploiting genuine quantum advantages, we propose a squeezed-light-enhanced phase noise feedback control system that fully leverages non-classical quantum resources. The core innovation lies in the strategy of conversion followed by suppression: an over-coupled three-mirror ring cavity operated in a half-detuned state is first employed to efficiently convert input phase noise into amplitude noise through its linear dispersive response near half-resonance; subsequently, a quantum amplitude squeezed state is injected into the vacuum port of the beam splitter within the classical feedback loop, thereby replacing the conventional vacuum noise source with a lower-noise squeezed vacuum. Theoretical analysis, based on a fully linearized quantum noise model, reveals that the injected squeezed light selectively and effectively suppresses the vacuum noise introduced by the beam splitter, which constitutes the dominant noise source in the out-of-loop detection field. Numerical results, obtained with realistic experimental parameters such as impedance-matching factor, beam-splitter transmittance, photodetection efficiency, and total system efficiency after losses, demonstrate that under optimal classical feedback gain, the out-of-loop noise variance reaches its minimum but remains strictly limited by the shot noise limit. In contrast, by introducing a -10 dB amplitude squeezed light which accounts for propagation and detection losses, the out-of-loop system surpasses the classical limit, achieving a significant additional noise reduction of 8.97 dB at a specific high-gain point to reach 18.46 dB total suppression, and pushing the overall phase noise suppression level further down to 18.68 dB at even higher gain regimes. Notably, the squeezed light contributes only a marginal 0.44 dB improvement to the in-loop field, highlighting that the primary quantum enhancement occurs in the out-of-loop observable. In conclusion, this study mathematically and physically proves that squeezed light can significantly enhance phase noise suppression even under limited probe light power conditions, where classical methods alone are insufficient. The proposed scheme provides a robust theoretical foundation, reveals the intrinsic mechanism linking squeezed-state properties to improved suppression performance, and offers a practical pathway toward approaching the shot noise limit for phase stabilization. This approach holds strong potential for next-generation high-sensitivity instruments, potentially enabling sensitivity gains beyond current thermal-noise-limited systems in applications such as gravitational wave detection and optical atomic clocks.