I noticed this same behaviour (Word 2013 "slow to save") on a new one page document.
Elsewhere, Anula D'Souza suggested:
"If you are able to work with Word without any problem then the issue might be due to add-ins.
In order to disable the add-ins follow the steps given below:
o In Word, click on File< Options< Add-ins.
o Under Manage select COM Add-ins.
o Click on GO options beside Manage, uncheck all the add-ins and click Ok.
o Try to open Word and check if it works fine."
Thanks Anula! On my system (Win 7 x64), under [COM Add-ins] I found
[x] Visual Studio tools for ...
were enabled. Unchecking this option seems to have fixed my (slow to save) problem.