After repeatedly working on a file, hitting save, attaching the file to an email, and realizing I sent the unsaved version of the file; I decided to dig into this odd behavior of Microsoft Word 2013.
My initial workaround was completely closing all Word windows, but I didn’t want to have to close all of my documents to save one.
After a little more digging a simple setting solves this problem. Go to File -> Options -> Advanced. Scroll down to Save and uncheck "Allow background saves".