Unfortunately, I only have Visio 2010 and Word 2010, but I hope some of this translates to Word 2013 decently.
First off, instead of copying and pasting the Visio diagram from Visio into Word, have you tried dragging and dropping the Visio diagram file itself into the Word document?
Next, I would try to embed the existing Visio diagram into your document. Insert > Object > Create from File > Browse to file > Check "Link to File" > "OK"
If you're looking to make the file such that your co-workers can edit within Word when Visio is installed, you may need to create a new Visio embedded object within Word. Try going to Insert > Object > Create New > Microsoft Visio Drawing > "OK". Then you will need to copy and paste your existing Visio diagrams into the newly created embedded object.
Hopefully one of these will get the functionality you're looking for.