I had some troubles with the selected answer. However, it worked when I added this line to .emacs:
(add-to-list 'tramp-default-proxies-alist '(".*""\\`root\\'""/ssh:%h:"))And then executed the following:
/sudo:ssh-host:file-on-ssh-host
It was slightly confusing because at one point I was prompted for the "root" password, but entering my user's password granted me access. It also universally works on all hosts on the network. Also, I can still do this to not be root:
/ssh:ssh-host:file-on-ssh-host