When a flexible link manipulator lacks a joint motor, how to use the remaining motors to achieve the control objective is a challenge, and the research in this direction is limited. This article presents a tip position control and vibration suppression approach for a planar two-link rigid-flexible (TLRF) underactuated manipulator with a passive first joint. First, we establish a dynamic model of the system by using the assumed mode method (AMM) and the Lagrangian modeling method. Then, we obtain the dynamic coupling relationship of the two links based on the dynamic model. According to this dynamic coupling relationship, we find that the passive rigid link can be controlled indirectly by controlling the active flexible link. Thus, we calculate the target angles of the two links by using the inverse kinematic method and design a controller for the active flexible link to stabilize it at its target angle and to suppress its vibration. Next, we optimize the parameters of this controller by using the genetic algorithm (GA). GA helps us simultaneously stabilize the passive rigid link at its target angle while realizing the control objective of the active flexible link. The simulation results demonstrate the effectiveness of the proposed control approach.