What the support link isn't clear about is that you have to edit the Link Widget itself which is under Appearance. (I'm using theme 2010 so your Dashboard may not look the same.)
So on your Dashboard you have to scroll down to Appearance, select Widgets, and go to the location of the Links Widget, in my case it's in Secondary Widget Area. There is the Links widget. Click 'edit' and the menu options will appear.
Then you can select the order in which they appear. The default is random. If you want to have them appear in a specific order you'll have to assign them a rank, select Rating and select that as your order. Otherwise you do have the option of having them alphabetized. They'll be alphabetized by the name you give them. Hope this helps. Took me a while to figure out - as in going on over an hour.
If you want me to send you screen shots of how I managed it I'll be happy to do so.