Saving and publishing changes to a prevailing measure works a bit differently than changes made to the narrative section of a goal page.
When the goal page is private/unpublished, any changes made to the prevailing measure also remain unpublished.
However, when a goal page is public-facing (i.e. has been published), changes made to the prevailing measure will immediately be reflected on the public-facing goal page. For example, if the author has the following published goal, for which the end date was December 2016 ...
... and the author wants to revise the goal for the following year like this:
... saving this change to the prevailing measure — moving the end date from December 2016 to December 2017 — will immediately be reflected in the public-facing version of the page.
Changes to the narrative section of a public-facing goal will be held in draft mode until the goal author explicitly promotes those changes to overwrite the public version.
For more details on creating and editing goals in the new experience, see the New Goal Editing Experience section of the support site.