This paper introduces a time-varying damped least squares (TV-DLS) enhanced recurrent neural network (RNN) control framework to achieve stable and precise trajectory tracking for robotic manipulators with unknown inertial parameters. A novel piecewise exponential damping function is proposed, smoothly transitioning from a high initial value to a minimal steady-state value, effectively mitigating transient joint acceleration spikes while preserving tracking accuracy. The approach integrates RNN-based online mass matrix estimation to compensate for dynamic uncertainties, with stability guaranteed by Lyapunov-based analysis demonstrating global exponential convergence. Simulations on a 7-DOF Franka Emika Panda manipulator reveal a significant reduction in initial acceleration peaks compared to static low-damping methods, alongside a steady-state position error of \(10^{-3}\) m. Unlike conventional static damping strategies, which struggle with the stability-precision trade-off, the proposed TV-DLS method dynamically adapts to operational phases, offering a robust solution for robotic control under model uncertainty. Comparative results highlight its superiority in balancing transient stability and steady-state performance, paving the way for advanced manipulator applications.