Considerable compliance of the lengthy oil-well drill string combined with low tool inertia and stick-slip friction between the tool and the rock bed causes notable undesired torsional vibrations of the drill-string electrical drive. In order to attenuate the torsional vibrations and, thus, improve both the quality and the productivity of the drilling process, an automatically-tuned active damping control strategy based on drill-string torque estimation is proposed in the paper. The core of the strategy includes a proportional-integral (PI) controller extended with estimator-based drill-string torque feedback loop. Furthermore, an appropriate algorithm that prevents drill-string back-spinning caused by the limited braking power of the power converter is presented. Finally, an auto-tuning algorithm is proposed, which is built around an adaptive Kalman filter-based estimator of drill-string drive natural frequencies. The drill-string control strategy is verified experimentally using a drill string hardware-in-the-loop setup under the laboratory conditions, as well as on an actual oil-drilling rig.