Bettas (and other anabatoids... hope I spelled it right) use both oxygen from the water and the air.. they do have gills (they are still fish after all ).
Gouramis and bettas, are both anabantoids. They have a organ called a labyrinth that makes them able to breath air. They will suffocate if they can't get to the surface.
I read a book about that anabatoids thing. The labrynith is a maze-like respatory organ located behind their mouth (the same way you can breath out of your mouth.)