I did this a couple of weeks ago but forgot to come back and post. I found a solution - via google, not my own solution - that works by uploading relevant Awstats files and then changing a path, much like what Tony described above.
Download Awstats from awstats.sourceforge.net
Upload cgi-bin (this can be renamed after) and icon folders in the download to root of your site
Change awredir.pl and awstats.pl in the folder uploaded to 755
Go to temp/awstats in cpanel and download the conf file for your domain from there (awstats.yourdomain.com.conf)
Open it and change DirIcons value to DirIcons="/icon/" and upload it to the cgi-bin (or renamed) folder uploaded earlier
Awstats can now be accessed www.yourdomain.com/cgi-bin/awstats.pl?config=yourdomain.com
without needing cpanel login, be sure to use cpanel to password protect the cgi-bin (or renamed) folder.