The micro-turbojet engine (MTE) is especially suitable for unmanned aerial vehicles (UAVs). Because the rotor speed is proportional to the thrust force, the accurate speed tracking control is indispensable for MTE. Thanks to its simplicity, the proportional–integral–derivative (PID) controller is commonly used for rotor speed regulation. However, the PID controller cannot guarantee superior performance over the entire operation range due to the time-variance and strong nonlinearity of MTE. The gain scheduling approach using a family of linear controllers is recognized as an efficient alternative, but such a solution heavily relies on the model sets and pre-knowledge. To tackle such challenges, a single neural adaptive PID (SNA-PID) controller is proposed herein for rotor speed control. The new controller featuring with a single-neuron network is able to adaptively tune the gains (weights) online. The simple structure of the controller reduces the computational load and facilitates the algorithm implementation on low-cost hardware. Finally, the proposed controller is validated by numerical simulations and experiments on the MTE in laboratory conditions, and the results show that the proposed controller achieves remarkable effectiveness for speed tracking control. In comparison with the PID controller, the proposed controller yields 54% and 66% reductions on static tracking error under two typical cases.