The short answer is that yes, you are responsible for making the FICA taxes whole. Note that you're not being charged anything you weren't already owing; you in fact got to have some of the money ahead of time, so you came out ahead in one sense. But technically, the portion of FICA tax that should be withheld is your responsibility to pay (and their responsibility to withhold).
Now, if you're living paycheck to paycheck this probably doesn't help you feel better about it, and since they didn't find it until the last month of the year, you end up being more short (they're probably trying to catch up by end of year, since each year's tax is separate to some extent).
If this is a burden that is unreasonable, you might ask either your direct supervisor or HR if there is anything they can do to help - whether it is spread the extra withholding out over a longer period of time, or give you an advance on some of your pay, or even just allow you to work some extra hours to make up for it temporarily. There may be many options to help you out here, and since it was their error initially they ought to be at least somewhat willing to work with you.
Also - as noted in comments below, you may see if you can lower your W4 withholding (income tax). Particularly if you are likely to get a refund, you could lower your withholding for the next few pay checks to shift some of that refund to now. Your employer may not be able to do that instantly, but it may be worth asking.