As a critical fluid-conveying component in mechanical structures, the stability and safety of fluid-conveying pipes during operation are of particular importance. Taking a fluid-conveying pipe as an example, this paper investigates the passive boundary control of the pipe by coupling an Inertial nonlinear energy sink at its elastic boundary. Firstly, the nonlinear control equations of a fluid-conveying pipe with elastic supports coupled with an inertial-based nonlinear energy sink are derived using the generalized Hamilton’s principle. Subsequently, the natural frequencies and mode shapes of the fluid-conveying pipe are obtained. Then, the control equations are discretized using the Galerkin truncation method, and the steady-state response of the coupled system is solved through numerical simulation based on the Runge-Kutta method. Finally, the influence of key parameters of the vibration absorber on the vibration reduction effect of the structure is discussed. The results indicate that the coupled inertial nonlinear energy sink exhibits effective vibration control for the fluid-conveying pipe without altering its inherent dynamic characteristics. The parameters of the inertial-based nonlinear energy sink have different effects on the vibration suppression performance: there exists an optimal damping coefficient, while increasing the inertance and the cubic nonlinearity enhances the vibration mitigation.