This paper aims to tackle the landing control issue of carrier-based unmanned aerial vehicle (UAV) in the presence of carrier air-wake and carrier deck motion. Firstly, a control method with finite-time convergence ability is designed for carrier-based UAV to achieve faster trajectory convergence and higher control accuracy. Theoretical analysis is conducted to demonstrate that the control method has asymptotic stability and finite-time error convergence ability. Subsequently, utilizing this control approach, an automatic carrier landing system (ACLS) is established, which consists of the guidance law subsystem, glide trajectory generation subsystem, flight control subsystem, auto-throttle control system, and the flight dynamic model of the aircraft. Then, by introducing the PID method for comparison, numerical simulation experiments are conducted. The experimental outcomes illustrate that in contrast to the PID approach, the NTSMC method exhibits better trajectory tracking performance and can effectively achieve precise and safe landing control even under the influence of environmental disturbances.