Add a docvariable field at the end of the document {DocVariable varDate}
Use the following macro to update the field.and save the document.
By naming it FileSave it will intercept Word's built-in Save command.
Sub FileSave()
Dim oVars As Variables
Dim oField As Field
Set oVars = ActiveDocument.Variables
oVars("varDate").Value = Format(Date, "dd/MM/yyyy")
For Each oField In ActiveDocument.Fields
If oField.Type = wdFieldDocVariable Then
oField.Update
End If
Next oField
On Error Resume Next
ActiveDocument.Save
End Sub