Interactions among an inerter, spring, and energy dissipation element (EDE) in an inerter system can result in a higher energy dissipation efficiency compared to a single identical EDE, which is referred to as the damping enhancement effect. Previous studies have mainly concentrated on the vibration mitigation effect of the inerter system without an explicit consideration or utilization of the damping enhancement mechanism. In this study, the theoretical essence of the damping enhancement effect is discovered, and a universal design principle is proposed for an inerter system. A fundamental equation is found and demonstrated on the basis of closed-form stochastic responses, which establishes a bridge between the damping deformation enhancement factor (DDEF) and the response mitigation ratio, thus clarifying the relationship of the damping enhancement effect and the response mitigation effect. Inspired by the equation, a novel damping-enhancement-based strategy is proposed to determine the key parameters of an inerter system. Following the performance-demand-based design philosophy, the parameters of the inerter system can be determined in the design condition of a target-damping-enhancement effect. Through the implementation of the damping enhancement equation, the damping parameter of an inerter system can be directly obtained by the prespecified DDEF and the displacement response mitigation ratio. The influence of parameters on the response mitigation effect and the damping enhancement effect is then investigated to determine ways of obtaining the other two parameters in an inerter system. Finally, design examples are conducted to verify the proposed strategy and the theoretical relationship revealed by the damping enhancement equation. The results show that the proposed design strategy explicitly utilizes the damping enhancement effect for vibration control, where the target of the DDEF is effective in enhancing the efficiency of the EDE for energy dissipation. In the design condition of the target DDEF, the i...