Can I Use Word 2013's Non-English Character Shortcuts System-Wide?

Have you tried enabling US-International English keyboard layout under Control Panel (instructions are for XP, Vista, and 7)?

This should work for all programs in Windows and according to the description below it has a similar input method as Microsoft Word that you are looking for. This site gives a description:

The US-International English keyboard layout is installed as above (or see installation details) and has these features:

It uses the following intuitive methods which work with most (or all) Windows applications, while keeping the familiar QWERTY keyboard.

Press one of the five modifier keys ` ' " ~ ^ , then the letter to be modified. ( ' then a = á, " then u = ü, ' then c = ç, etc. ). Press the right alt key + another key. Example: right alt + , = ç (or ' + c)