Wire-harnessing tasks pose great challenges to be automated by the robot due to the complex dynamics and unpredictable behavior of the deformable wire. Traditional methods, often reliant on dual-robot arms or tactile sensing, face limitations in adaptability, cost, and scalability. This paper introduces a novel single-robot wire-harnessing pipeline that leverages a robot's twisting motion to generate necessary wire tension for precise insertion into clamps, using only one robot arm with an integrated force/torque (F/T) sensor. Benefiting from this design, the single robot arm can efficiently apply tension for wire routing and insertion into clamps in a narrow space. Our approach is structured around four principal components: a Model Predictive Control (MPC) based on the Koopman operator for tension tracking and wire following, a motion planner for sequencing harnessing waypoints, a suite of insertion primitives for clamp engagement, and a fix-point switching mechanism for wire constraint updating. Evaluated on an industrial-level wire harnessing task, our method demonstrated superior performance and reliability over conventional approaches, efficiently handling both single and multiple wire configurations with high success rates.