Compared with stochastic differential equations (SDEs) driven by white noise, random differential equations (RDEs) generated by colored noise are claimed to be more practical. This article considers reduced-order high-gain observer (ROHGO)-based neural tracking control on random nonlinear systems having output delay. In order to foster the design and analysis, the estimated states and the estimation errors are scaled by the high gain of the observer. Based on neural network (NN) approximation and state observation, an adaptive controller is designed for the overall system using the backstepping method. It is proved that all the closed-loop signals are bounded almost surely, letting alone the tracking error. By tuning the related design parameters, the asymptotic tracking error could be regulated arbitrarily small. Within the best of our knowledge, this article serves as the first attempt for NN-based control on RDE systems. Finally, the validity of main results is confirmed by a simulation example.