To address the stability issues of Model Predictive Path Integral (MPPI) control in constrained vehicle waypoint tracking, this paper proposes a Lyapunov-stabilized MPPI (LS-MPPI) strategy with closed-loop asymptotic stability guarantees. First, a unified quadratic continuous barrier cost function is constructed to resolve the issues of sampling gradient sparsity and weight degradation caused by traditional hard constraint penalties. Second, a dynamic non-increasing arbitration mechanism between stochastic sampling and warm-start paths is introduced to ensure the recursive feasibility. Furthermore, sufficient conditions for closed-loop asymptotic stability are established via terminal cost functions and Lyapunov’s arugument. Finally, multi-scenario simulation results demonstrate that, compared to standard MPPI, the proposed strategy reduces the lateral and heading root mean square errors (RMSE) by 43.5% and 23.0%, respectively.