The purpose of this paper is to introduce a novel adaptive neural network-based control scheme for the Furuta pendulum, which is a two degree-of-freedom underactuated system. Adaptation laws for the input and output weights are also provided. The proposed controller is able to guarantee tracking of a reference signal for the arm while the pendulum remains in the upright position. The key aspect of the derivation of the controller is the definition of an output function that depends on the position and velocity errors. The internal and external dynamics are rigorously analyzed, thereby proving the uniform ultimate boundedness of the error trajectories. By using real-time experiments, the new scheme is compared with other control methodologies, therein demonstrating the improved performance of the proposed adaptive algorithm.