A significant increase of system complexity and state changes requires an effective data-driven system identification and machine learning algorithm to deal with the control of nonlinear systems. Using streams of data collected from the system, the data-driven controller aims to stabilize the unknown nonlinear systems with modeling uncertainties and external disturbances. The paper proposes a novel data-driven adaptive control approach with the backstepping strategy for online control of unknown nonlinear systems with no human intervention. A new meta-cognitive fuzzy-neural model is first introduced to construct the unknown system dynamics and utilize the self-adaptive tracking error as the learning parameters to determine the deletion of the state data, adapt the structure and parameters of the controller using the information extracted from nonstationary data streams. Subsequently, the control law is constructed based on the meta-cognitive fuzzy neural model rather than the actual systems and the backstepping control strategy. Then, the stability analysis of the closed-loop system is presented from the Lyapunov function and shows that the tracking errors converge to zero. In the proposed control scheme, the bound of the control input is considered and ensured via the stable projection-type adaptation laws of the parameters. Moreover, in order to further save online computation time, only the parameters of the rule nearest to the current state are updated while those of other rules maintain unchanged. This is different from the existing studies where the parameters of all rules are updated. Finally, various simulation results from an inverted pendulum system and a thrust active magnetic bearing system demonstrate the superior performance of the proposed meta-cognitive fuzzy-neural control approach.