Due to the inherent deformability of continuum robots (CRs), closed-loop control strategies of CRs come with some challenges, including sensor integration within a limited dimension and uncertainties with the CR system. In this paper, to obtain the spatial shape information and tip position of a cable-driven CR, a draw tower grating (DTG) sensor is integrated. A shape sensing algorithm is proposed to reconstruct the shape of the CR based on the relation between the DTG sensor and the CR centerline. Then, with the feedback information of the DTG sensor, a damped least-squares (DLS)-based closed-loop controller is implemented to improve the trajectory tracking performance of the CR. To verify the effectiveness of the proposed scheme, comparative experiments are performed.