Change Master Page via the UI in a Non-Publishing Site

Any sites without the Publishing features activated will be missing the Master Page link in the Look and Feel section in Site Settings:

vs with the Publishing features activated:

If you need to change the master page on a site that does not have the publishing features activated you can just add the following to the url of your site:

/_layouts/ChangeSiteMasterPage.aspx

and hit that directly.  All the functionality will still work.