It sounds like you're saving a copy to the desktop (using the Save as... command) and then deleting the copy.
You need to figure out where the original file is. There are three ways of doing this:
1. Pay attention when you're saving it in the first place...when you're saving it, look at the dialog that you enter the file name. It sounds like it's a little too late for this, so....
2. Use Spotlight, the search function in the right-hand side of the menu bar, with the magnifying-glass icon. Just enter the name of the file, wait until it shows up in the list, and then click Show All. The Spotlight window will open, and you'll be able to move the file to the Trash, or anywhere else.
3. Command-click on the title icon. If you can open the document in Word, hold down the Command key (the key on either side of the spacebar) and click on the icon next to the title at the top of the document window. It will display a menu that shows the folder that contains the document (and the folders that contain that folder, and so on.)
Once you've figured out where the original document is, you can Trash it permanently.