We develop and experimentally test a prescribed performance control methodology for trajectory tracking of underactuated autonomous aerial vehicles under unknown time-varying disturbances. An extended state observer is first proposed to filter position and velocity measurements, and to compensate for disturbances, including model parametric uncertainty and/or exogenous perturbations, e.g., wind. Then, by resorting to a barrier function and to constraints on the transient and steady-state response of the position error, we derive a thrust command law that follows from a desired stabilizing force. This force is inherently associated with the vehicle's attitude, which is controlled using angular velocity commands. The errors stemming from the closed-loop system are shown to be uniformly ultimately bounded. Simulation results, including comparison data, are presented to validate the proposed solution. Experimental results are also included to showcase the performance of our strategy in a real-world application using a quadrotor vehicle.