It is possible to get a late fee waived on your credit card by calling customer service at the number you see on the back of your card and explaining the situation. This is much more likely to work if you have a great payment history, but even if not, there's no harm in asking.
As rule of thumb, credit card companies are less flexible with cardholders who have a history of missing monthly payments. To avoid getting hit with late fees, make sure to pay at least your minimum required payment each month. The best way to ensure that you never miss a payment is by setting up autopay.