This article presents an intelligent control strategy for the position and attitude tracking of a quadrotor using a nonsingular fast-terminal sliding mode and disturbance observer. The quadrotor system is divided into an underactuated position and the fully-actuated attitude subsystem. A single hidden layer feed-forward neural network is used for the estimation of unknown dynamics associated with the attitude subsystem. Moreover, a nonlinear disturbance observer is designed to tackle the unknown bounded external disturbances acting on the attitude subsystem by supplying its estimation in the control laws that help in mitigating the chattering phenomena. The proposed methodology ensures the finite-time convergence of the tracking error and does not prone to the singularity problem in control. The closed-loop finite-time stability is investigated using the Lyapunov theory. Besides, an expression for the convergence time has been derived. The effectiveness of the proposed scheme is initially assessed using numerical simulations for the way-point tracking as well as circle tracking tasks and then validated in real-time using DJI Matrice 100 by performing a lemniscate tracking task. To show the superiority of the proposed methodology over existing methods, a comparative study has been done and found the improved tracking performance in the proposed approach.