Self-sustaining process (SSP) (Waleffe, 1997) is normally studied in parallel shear flows or in the subcritical regime of Taylor-Couette flow (TCF). Compared with shear-driven flows, the self-sustainment of exact coherent structures in TCF is more involved because of the coexistence of shear and centrifugal instabilities. We study a series of self-sustained drifting-rotating waves (DRW) underlying spiral turbulence (SPT) with rich dynamical behaviors. The control equations are the incompressible Navier-Stokes equations of the perturbation field. Solenoidal Petrov-Galerkin scheme formerly formulated by Meseguer et al. (2007) is adapted to annular-parallelogram domains for discretization. And the solenoidal velocity perturbation is approximated through a Fourier-Chebyshev spectral expansion. Forth order linearly implicit time scheme is chosen for DNS computations. Newton-Krylov (Kelley, 2003) pseudo-arclength continuation, along with Arnoldi eigenvalue methods are used to track and monitor the exact coherent structures and their linear stability, respectively. We have reported a family of high subcritical three-dimensional nonlinear wave solutions that spread over both subcritical and supercritical regimes of counter-rotating Taylor-Couette flow. In the subcritical regime, the SSP is proved to be the reason for the self-sustainment of DRW. In the supercritical regime, where SPT exists, the dynamical relevance of the DRW within SPT dynamics is shown by replicating the wave in the azimuthal direction so that it fills in a narrow parallelogram domain revolving around the apparatus perimeter. Results show that the self-sustained vortices eventually concentrate into a localized pattern which satisfactorily reproduces the topology and properties of the SPT calculated in a large periodic domain of sufficient aspect ratio that is representative of the real system.