First, congratulations for paying down your credit card balance. In this economy, most of us find that a tough achievement. However, you've already noticed a quirk of modern accounting: a final finance charge representing interest on your balance during your previous statement cycle. So, yes, you do have to pay the amount. In fact, losing track of finance charges after a payoff has become one of the main reasons Americans wind up in trouble with collections agencies.

Most credit cards offer a "grace period," ranging from three to five weeks. During that grace period, you usually won't incur finance charges on new purchases. That's what makes rewards credit cards so attractive for consumers who might otherwise pay cash for everything. Just rack up your monthly purchases on that account, then pay your bill in full the very moment you get your statement. You earn rewards points or cash back, without paying interest.

Once that statement's in your hands (or in your digital inbox), you're paying interest on the daily balance of all the purchases that have fallen out of your grace period. Even if you pay the total amount on your statement and never make another charge on that account, that tiny finance charge will lie in wait until next month's bill.

Ignore or miss that payment, and you could wind up with significant late fees and collections activity over a few dollars in interest. Under the terms and conditions for most popular credit cards, banks can charge you $35 or more for each month you fail to clear that balance. Those fees earn interest of their own, until a bank charges off the account and sends it to a third party collections agency.

You're not the only one with this problem, which often crops up after processing a balance transfer offer. According to statistics compiled by the Consumer Finance Protection Bureau, 6 percent of credit card complaints stemmed from problems with closing an account. Two percent of the CFPB's credit card complaints involved the payoff process itself.

Because of the way most banks' systems operate, a customer service agent might not even see that you owe this amount if you call for confirmation that you zeroed out your balance. The best way for you to ensure you've totally cleared your debt is to wait until you've received a final statement from your bank with an amount due of $0.00. Keep a copy of that statement on file, in case of any future dispute.

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