You should be able to log in as root, because usually a percentage of the partition's size is reserved in order to always enable root login for rescue operations and such. See this U&L Q&A:
Reserved space for root on a filesystem - why?
What you won't be able to do, however, is log in as a regular user from your display manager then switch to root or use
sudo from a shell in a terminal.
You have two alternatives instead:
- Switch to a VT (press Ctrl + Alt + F2, for example), log in as root from there and free some space.
- At boot time opt for single user mode to get a root shell that would also help you free some space for your regular login.
This assumes that the reason your partition was filled up is due to regular user activity and not activity by root processes. In such cases, you might need to resort to mounting the partition on a Live system and freeing up the space from there. Thanks to Alexios' comment for bringing this up.