Navigation underwater traditionally is done by keeping a safe distance from obstacles, resulting in "fly-overs" of the area of interest. An Autonomous Underwater Vehicle (AUV) moving through a cluttered space, such as a shipwreck, or a decorated cave is an extremely challenging problem and has not been addressed in the past. This paper proposed a novel navigation framework utilizing an enhanced version of Trajopt for fast 3D path-optimization with near-optimal guarantees for AUVs. A sampling based correction procedure ensures that the planning is not limited by local minima, enabling navigation through narrow spaces. The method is shown, both on simulation and in-pool experiments, to be fast enough to enable real-time autonomous navigation for an Aqua2 AUV with strong safety guarantees.