Here's how I solved the problem :

1. Clean any saved password you have for the affected website.
2. Install the "Saved Password Editor 2.7.1" addon.
3. Have Firefox save the wrong password once.
4. Go into the addon's personalized password manager.
5. Edit the password and save the popup as an image for future reference.
6. Clean any saved password you have for the affected website (again).
7. Go to the website, log in and this time, refuse to save the password. This will put the website on the exception list.
8. Go into the addon's personalized password manager.
9. Create a new saved password and enter the informations as stated in the printscreen from previous step, but input the right password.

MAGIC! Firefox will auto complete the website's information AND will not ask for an update to the password.

It's a shady workaround for an option that should be included, but it works!

Enjoy!