The solution was simple to do, although desperately hard to find - hence its appearence here:
Login to the server as root and edit the /etc/nginx/fastcgi_params file:
Simply comment out (add # at the beginning of the line) to the line that says:
fastcgi_param SCRIPT_NAME $fastcgi_script_name;
Now restart nginx and you're done!
[UPDATE: you may need to check the lines in your configure.php file from the admin directory - they need to point correctly, without using the PHP_SCRIPT line:
define('DIR_WS_ADMIN', '/#your admin folder#/');
define('DIR_WS_CATALOG', '/');
define('DIR_WS_HTTPS_ADMIN', '/#your admin folder#/');
[UPDATE: you may need to check the lines in your configure.php file from the admin directory - they need to point correctly, without using the PHP_SCRIPT line:
define('DIR_WS_ADMIN', '/#your admin folder#/');
define('DIR_WS_CATALOG', '/');
define('DIR_WS_HTTPS_ADMIN', '/#your admin folder#/');