The manipulation of deformable linear flexures has a wide range of applications in industry, such as cable routing in automotive manufacturing and textile production. Cable routing, as a complex multi-stage robot manipulation scenario, is a challenging task for robot automation. Common parallel two-finger grippers have the risk of over-squeezing and over-tension when grasping and guiding cables. In this paper, a novel eagle-inspired fingernail is designed and mounted on the gripper fingers, which helps with cable grasping on planar surfaces and in-hand cable guiding operations. Then we present a single-grasp end-to-end 3D cable routing framework utilizing the proposed fingernails, instead of the common pick-and-place strategy. Continuous control is achieved to efficiently manipulate cables through vision-based state estimation of task configurations and offline trajectory planning based on motion primitives. We evaluate the effectiveness of the proposed framework with a variety of cables and channel slots, significantly outperforming the pick-and-place manipulation process under equivalent perceptual conditions. Our reconfigurable task setting and the proposed framework provide a reference for future cable routing manipulations in 3D space.