This paper deals with the use of a passive hydrophone array (PHA) for following non-stationary underwater objects with a miniature biomimetic autonomous underwater vehicle. Current acoustic underwater localization systems are large and expensive. They are not well suited for using on miniature, low-cost underwater vehicles. However, compact size and low cost are necessary when developing autonomous vehicles that can be widely used by recreational and professional divers. As an alternative, a simple hydrophone array is developed for a biomimetic 4-flipper U-CAT AUV (Autonomous Underwater Vehicle). The array is used to measure the bearing of a single acoustic beacon. Two control approaches are proposed for navigating the vehicle towards the beacon. The whole system is validated both in a simulator and in a natural environment using two scenarios: 1) navigating towards a stationary beacon and 2) navigating towards a beacon mounted on a diver. The results show that the proposed control system is a viable approach to be used for diver following.