For the single file edition of TW5, the only way of saving changes under Google Chrome is to use the fallback HTML download saver - which is to say that clicking the "save/download" button will download a fresh copy of the HTML file. That's clumsy enough that most people wouldn't want to do it except in an emergency.
Sadly, a simple Chrome Extension can't access the file system outside of a sandbox, so it won't be possible to build something as straightforward as TiddlyFox for Google Chrome. That may change as Chrome matures.
So, right now, if you're unable to use Firefox, and need to use Chrome, you're probably best off using TiddlyWiki5 with the node.js server edition, or via TiddlySpace/TiddlyWeb. (Integration with the latter isn't quite ready for primetime, however).
Best wishes
Jeremy