They'll probably do it. Just tell them you've been with them a long time and only had one other late payment. Tell them why you're late, even if it's just that you forgot or tell them you thought because you paid online it would post that same day but still admit fault. Tell them you'll be extremely careful in the future and ask if they'd be willing to waive the fee as a "one time customer service."
Just be nice, odds are that they'll do it. But, don't do it today of course. They'll just tell you to pay it ontime with their express option. Wait till you get the next bill.
You might also want to consider paying the express fee. Is it less than $39? It doesn't make sense to cross your fingers to get $39 waived when $12 would have avoided it in the first place, not worth the gamble.
I haven't seen any other answers to your question but, just in case, don't worry about your credit. Late payments aren't reported to your credit unless they are over 30 days late, meaning you've completely missed a payment--that can happen even if you plan on "paying double" the next month like some people think works, it doesn't.