Our little princess has never flown coach. Too bad dogs don't accrue miles, she'd be at least a GM by now.
Call DL. Tell them you're booking a pet in cabin - there's a $125 fee, which you pay at check-in (IIRC, you can't do OLCI when traveling with a pet).
http://www.delta.com/content/www/en_...ons.html#carry