Injection molding is an important process to produce goods made of plastics. In this process, the polymer granulate is melted and injected into a cooled form. The cyclic injection process consists of the following phases: filling, packing and cooling, plastication, and ejection of the final part. In this work, a model-based optimal control strategy for the filling and packing phase is developed. During the filling phase, the screw is moved forward by means of two hydraulic cylinders, which injects the molten polymer into the form. A non-return valve prevents backflow of the polymer via the barrel. In the packing phase, a certain pressure-profile is applied to the melt to compensate for shrinking of the polymer during the solidification. The control tasks of the two phases can be summarized as follows: In the filling phase, the movement of the screw has to be controlled according to a user-defined velocity trajectory. In general, the velocity profile is given as a function of the current screw position. During the filling phase, the system has to be kept below a certain pressure limit. In the subsequent packing phase, a desired pressure-profile is to be tracked as a function of time. In both phases, all the limits of the drive speed, drive acceleration, and drive torque have to be respected. The controller has to follow the desired trajectories as good as possible while keeping the injection unit within the given limits. This control task is solved by a nonlinear model-predictive control, which allows for an optimal control performance independent of the operating point of the injection unit. The proposed control concept is implemented on a rapid-prototyping platform and experimentally validated on an industrial injection molding machine.