One of the main limiting factors in deployment of marine robots is the issue of energy sustainability. This is particularly challenging for traditional propeller-driven autonomous underwater vehicles which operate using energy intensive thrusters. One emerging technology to enable persistent performance is the use of autonomous recharging and retasking through underwater docking stations. This paper presents an integrated navigational algorithm to facilitate reliable underwater docking of autonomous underwater vehicles. Specifically, the algorithm dynamically re-plans Dubins paths to create an efficient trajectory from the current vehicle position through approach into terminal homing. The path is followed using integral line of sight control until handoff to the terminal homing method. A light tracking algorithm drives the vehicle from the handoff location into the dock. In experimental testing using an Oceanserver Iver3 and Bluefin SandShark, the approach phase reached the target handoff within 2 m in 48 of 48 tests. The terminal homing phase was capable of handling up to 5 m offsets with approximately 70% accuracy (12 of 17 tests). In the event of failed docking, a Dubins path is generated to efficiently drive the vehicle to re-attempt docking. The vehicle should be able to successfully dock in the majority of foreseeable scenarios when re-attempts are considered. This method, when combined with recent work on docking station design, intelligent cooperative path planning, underwater communication, and underwater power transfer, will enable true persistent undersea operation in the extremely dynamic ocean environment.