Combining safety objectives with stability objectives is crucial for safety-critical systems. Existing studies generally unified these two objectives by constructing Lyapunov-type barrier functions. However, insufficient analysis of key set relationships within the system may render the proposed safety and stability conditions conservative, and these studies also did not provide how to use such conditions to design safety-stability control strategies. This paper proposed a feasible and constructive design to achieve stabilization of safety-critical systems by a relaxed converse Lyapunov-barrier approach. By analyzing the relationships between a series of sets associated with the safety-critical system, the stability and safety conditions can be appropriately relaxed. Then, with the help of relaxed converse control Lyapunov-barrier functions (RCCLBFs), a theoretical result was obtained for the stability of affine nonlinear systems with safety constraints. Subsequently, a constructive method was developed for a second-order strict-feedback system to transform the process of solving RCCLBFs into a Lyapunov-like stabilization problem. Finally, the proposed safety-stability control method is exerted on a robotic system and demonstrated by simulations.