Instead of streaming the entire desktop (or a portion of it), you'll actually want to set up the ["Game Capture" source](https://obsproject.com/wiki/Sources-Gui

Instead of streaming the entire desktop (or a portion of it), you'll actually want to set up the ["Game Capture" source](https://obsproject.com/wiki/Sources-Guide#game-capture https://obsproject.com/wiki/Sources-Guide#game-capture), and configure it to automatically attach to any fullscreen application. Then add an audio stream, and you're good to go.