First, the method recommended by the official tutorial simply does not work and should be ignored. Luckily, a user named Wellspring provided a link in the comments section to a tutorial that actually works. Big thanks to him, and to Matei Cezar at TecMint.com for writing the article.
Basically, you enter these commands in this order. Read the linked tutorial for explanations.
$ sudo cp /etc/apt/sources.list /etc/apt/sources.list.bak
$ sudo cp /etc/apt/sources.list.d/official-package-repositories.list /etc/apt/sources.list.d/official-package-repositories.list.bak
$ sudo sed -i 's/saucy/trusty/' /etc/apt/sources.list
$ sudo sed -i 's/petra/qiana/' /etc/apt/sources.list
$ sudo sed -i 's/saucy/trusty/' /etc/apt/sources.list.d/official-package-repositories.list
$ sudo sed -i 's/petra/qiana/' /etc/apt/sources.list.d/official-package-repositories.list
## For Package Sources only if they are enabled ##
$ sudo sed -i 's/saucy/trusty/' /etc/apt/sources.list.d/official-source-repositories.list
$ sudo sed -i 's/petra/qiana/' /etc/apt/sources.list.d/official-source-repositories.list
## For Getdeb Sources only if they are enabled ##
$ sudo sed -i 's/saucy/trusty/' /etc/apt/sources.list.d/getdeb.list
$ sudo sed -i 's/petra/qiana/' /etc/apt/sources.list.d/getdeb.list
$ sudo apt-get update
$ sudo apt-get dist-upgrade
$ sudo apt-get upgrade
After dist-upgrade, enter "y" whenever prompted in the terminal, and answer "yes" and "install the package maintainer's version" when prompted in the graphical interface.
It's not mentioned in the tutorial, but I was also asked to restart xscreensaver and xlockmore while upgrading. This is probably unique to XFCE distributions and not all LM. To do this, open a new terminal and enter:
$ sudo killall xscreensaver
$ sudo killall xlockmore
Credit to hmagoo of UbuntuForums.org for this solution.
Then proceed as normal. The process takes a while, once it's done go ahead and restart. Again, I urge you to read the linked tutorial. This post is a stripped down version mainly for my own reference.