The post-buckling behavior and nonlinear vibration of a fluid-conveying pipe composed of a functionally graded material were analytically studied. The power-law material property was considered as continuously varying across the direction of the pipe wall thickness. A nonlinear governing equation for the pipe and relevant boundary conditions were derived based on Hamilton’s principle. The post-bucking configurations of the pipe were analytically predicted. The closed-form expression of the nonlinear free vibration of the pipe was determined using the homotopy analysis method. Numerical results are presented to display the dependence of the flow velocity, fluid density, and the initial stress on the post-buckling configurations. It was concluded that the statics and dynamics are significantly changed by the material properties, which suggests that the dynamic behavior of pipes may be tailored by use of man-made functionally graded materials.