In this paper a control strategy to perform the desired orientation of the end-effector in the path following task is presented. It is an extension of the traditional path following task definition for a manipulator as both end-effector position and orientation with respect to a path are taken into account. The path following algorithm is based on a parametric approach–a robot is described with respect to the Serret–Frenet frame associated with the desired path. The formalism of unit quaternions is used to define the relative orientation of the robot end-effector. Such an approach introduces additional constraints of the first order to be satisfied during the motion along the path. In the paper a control law for a redundant holonomic manipulator is proposed to perform the desired position and orientation with respect to the given path. The strategy shows how to deal with the redundant information given by the quaternions. Moreover, the definition of the desired state with respect to the path is proposed so that rapid motions of the manipulator are avoided in spite of significant initial errors, e.g. the robot is located outside the path in the initial state. The theoretical results are validated with a numerical analysis and experimental study. The achieved consistency of the results confirms the practical applicability of the proposed solution.