As noted above by Daniel… saving a page does not instantly generate a change to the sitemap. It’s done as a background task. It’s literally the same task that updates for the forum threads.
You don’t need to manually submit it to Google. They’ll crawl your site every so often looking for changes. It will never be instantly but the frequency is determined by Google based on the frequency of content updates, popularity of content, and a bunch of other things that is decided by Google.