Autonomous underwater navigation remains a challenging problem due to limited sensing capabilities and the difficulty of constructing accurate maps in underwater environments. In this paper, we propose a Diffusion-based Underwater Visual Navigation policy via knowledge-transferred depth features, named DUViN, which enables vision-based end-to-end 4-DoF motion control for underwater vehicles in unknown environments. DUViN guides the vehicle to avoid obstacles and maintain a safe and perception awareness altitude relative to the terrain without relying on pre-built maps. To address the difficulty of collecting large-scale underwater navigation datasets, we propose a method that ensures robust generalization under domain shifts from in-air to underwater environments by leveraging depth features and introducing a novel model transfer strategy. Specifically, our training framework consists of two phases: we first train the diffusion-based visual navigation policy on in-air datasets using a pre-trained depth feature extractor. Secondly, we retrain the extractor on an underwater depth estimation task and integrate the adapted extractor into the trained navigation policy from the first step. Experiments in both simulated and real-world underwater environments demonstrate the effectiveness and generalization of our approach. The experimental videos are available at https://www.youtube.com/playlist?list=PLqt2s-RyCf1gfXJgFzKjmwIqYhrP4I-7Y.