As an emerging class of continuum robots, tensegrity continuum robots (TCRs) utilize tension cables and compression members to achieve an optimal balance between compliance and mechanical robustness. Their locomotion involves large-scale rigid-body motions and elastic deformations, coupled with cable slack. The tip trajectory tracking control of such a system is challenging. This work proposed a differential-algebraic equations (DAEs) model-based instantaneous optimal control (IOC) framework for the tip’s position and orientation tracking of a TCR with slack cables. First, based on complementarity theory and Fischer-Burmeister function, the non-smooth problem of cable slack can be described as a continuous differentiable algebraic equation. Subsequently, combing with the dynamic differential equations derived via the position finite element method (PFEM), the control model of the TCR with rigid-flexible behaviors and non-smooth cable slack can be built as a DAEs system. Then, the continuous tip tracking problem is approximated to a series of IOC problems at each discrete time slot. Finally, considering the control input saturation constraints, a small-scale optimization problem is derived for solving these IOC problems. The method provides a novel and unified control framework for addressing the tip trajectory tracking issues of the TCR, and the non-smooth property of cable slack is naturally involved in the IOC controller. Numerical and Experimental results demonstrate that the proposed IOC controller is effective. The maximum relative error of the tip’s position is about 1.7%, and the root mean square error of the orientation angle is less than 3°.