Self-sustaining process (SSP) of coherent structures in the absence of linear instability of the laminar flow has been the topic for many scientific activities since the paper published by Waleffe (1997). Comparing to merely shear-driven flows, the self-sustainment of exact coherent structures in Taylor-Couette flow (TCF) is more involved because of the interplay of shear and rotation. Spiral turbulence (SPT) can persist in the supercritical regime of counter-rotating TCF, beyond the linear instability of the laminar circular Couette flow (Prigent et al. (2002); Meseguer et al. (2009)), meaning that both the shear and the centrifugal instabilities contribute to the generation of streamwise (azimuthal) vorticity. This fact has gone largely unnoticed in the literature, where the origin of the stripe can be solely explained by the stability of both the basic flow and the autonomous vortex emerged from the SSP. We report a self-sustained drifting-rotating wave (DRW) generated via a saddle-node bifurcation in the absence of linear instability of the base flow in counter-rotating Taylor-Couette flow (Wang et al., 2022). The DRW is captured in minimal computational box with two of whose sides aligned with the cylindrical helix described by the spiral pattern. Newton-Krylov continuation, along with Arnoldi eigenvalue methods are used to converge the DRW solutions, and monitor their linear stability, respectively. It is found that these DRW are linearly stable in a very narrow region of parameter space, close to the saddle-node bifurcation where they are created. The stable DRW solutions undergo a Hopf bifurcation, generating periodic solutions that eventually lead to a chaotic attractor via period doubling cascade. The SSP is proved to be at work in the subcritical regime, as roll and streak constituents of the DRW trigger the azimuthally-dependent wave component and are, at the same time, regenerated by it. We manage to track the DRW solutions up to the supercritical regime where the SPT exists. These DRW solutions are then replicated in the azimuthal direction so that they fill in a narrow parallelogram domain revolving around the apparatus perimeter. 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.